When planning a trip, knowl-

edge of a coordinate system is

helpful.

ALGEBRA, GRAPHS,
AND FUNCTIONS
lgebra, and, in particular, word problems: The very mention of them is enough

A to feighten many peopk, and yet '[geb" ;, one of the most practical tool, fa<
solving everyday problems. You probably use algebra in your daily life with-
out realizing it.
For example, you use a coordinate system when you consult your car map to find direc-
tions to a new destination. You solve simple equations when you change a recipe to increase
or decrease the number of servings. To evaluate how much interest you will earn on a sav-
ings account or to figure out how long it will take you to travel a given distance, you use
common formulas that are algebraic equations.
The symbolic language of algebra makes it an excellent tool for solving problems. Sym-
bolism has three advantages. First, it allows us to write lengthy expressions in compact form.
Second, symbolic language is clear-each symbol has a precise meaning. Finally, symbolism
allows us to consider a large or infinite number of separate cases with a common property.
The English philosopher Alfred North Whitehead explained the power of algebra when
he stated, "By relieving the brain of all unnecessary work, a good notation sets the mind
free to concentrate on more advanced problems and in effect increases the mental power of
the race."
 Algebra is a generalized form of arithmetic. The word algebra is derived from the
Arabic word al-jabr (meaning "reunion of broken parts"), which was the title of a
book written by the mathematician Muhammed ibn-Musa al Khwarizmi in about
A.D. 825.
Why study algebra? You can solve many problems in everyday life by using
arithmetic or by trial and error, but with a knowledge of algebra you can find the solu-
tions with less effort. You can solve other problems, like some we will present in this
chapter, only by using algebra.
Algebra uses letters of the alphabet called variables to represent numbers. Often
DID YOU KNOW the letters x and yare used to represent variables. However, any letter may be used as
a variable. A symbol that represents a specific quantity is called a constant.
Multiplication of numbers and variables may be represented in several different
ways in algebra. Since the "times" sign might be confused with the variable x, a dot
between two numbers or variables indicates multiplication. Thus, 3 . 4 means 3 times
4, and x . y means x times y. Placing two letters or a number and a Jetter next to one
another, with or without parentheses, also indicates multiplication. Thus, 3x means 3
times x, xy means x times y, and (x)(y) means x times y.
ALGEBRISTA An algebraic expression (or simply an expression) is a collection of variables,
numbers, parentheses, and operation symbols. Some examples of algebraic expres-
sions are

3x + I
~ 2x - 3'

SANGRADOR Two algebraic expressions joined by an equal sign form an equation. Some ex-
amples of equations are

magineyourself walking down a

I street in Spain during the Middle
Ages. You see a sign over a door:
The solution to an equation is the number or numbers that replace the vari-
"Algebrista y Sangrador." Inside,
able to make the equation a true statement. For example, the solution to the equa-
you would find a person more ready
to give you a haircut than help you tion x + 3 = 4 is x = 1. When we find the solution to an equation, we solve the
with your algebra. The sign trans- equation.
lates into "Bonesetter and Bloodlet- We can determine if any number i a solution to an equation by checking the so-
ter," relatively simple medical treat- lution. To check the solution, we substitute the number for the variable in the equa-
ments administered in barbershops tion. If the resulting statement is a true statement, that number is a solution to the
of the day. equation. If the resulting statement is a false statement, the number is not a solution to
The root word al-jabr, which the the equation. To check the number x = 1 in the equation x + 3 = 4, we do the
Muslims (Moors) brought to Spain following.
along with some concepts of alge-
bra, suggests the restoring of broken x+3=4
parts. The parts might be bones, or
+3=4
they might be mathematical expres-
sions that are broken into separate 4=4
parts and the parts moved from one
side of an equation to the other and The same number is obtained on both sides of the equal sign, so the solution is
reunited in such a way as to make a correct. For the equation x + 3 = 4, the only solution is x = 1. Any other value of x
solution more obvious. would result in the check being a false statement.
 To evaluate an expression means to find the value of the expression for a given
value of the variable. To evaluate expressions and solve equations, you must have an
understanding of exponents. Exponents (Section 5.6) are used to abbreviate repeated
multiplication. For example, the expressions 52 means 5 . 5. The 2 in the expression
52 is the exponent, and the 5 is the base. We read 52 as "5 to the second power" or "5
squared," and 52 means 5 . 5 or 25.
In general, the number b to the nth power, written bn, means

b·b·b·· .. ·b
~
n factors of b

An exponent refers only to its base. In the expression -52, the base is 5. In the
expression (-5)2, the base is -5.

-52 = _(5)2 = -1(5)2 = -1(5)(5) -25

(-5)2 = (-5)( -5) = 25

Order of Operations
To evaluate an expression or to check the solution to an equation, we need to know the
order of operations to follow. For example, suppose we want to evaluate the expres-
sion 2 + 3x when x = 4. Substituting 4 for x, we obtain 2 + 3·4. What is the value
of 2 + 3' 4? Does it equal 20, or does it equal 14? Some standard rules, called the
order of operations, have been developed to ensure that there is only one correct an-
swer. In mathematics, unless parentheses indicate otherwise, always perform multipli-
cation before addition. Thus, the correct answer is 14.

I. First, perform all operations within parentheses or other grouping symbols (ac-
cording to the following order).
2. Next, perform all exponential operations (that is, raising to powers or finding
roots).
3. Next, perform all multiplications and divisions from left to right.
4. Finally, perform all additions and subtractions from left to right.
DIDYOUKNOW TIMELY TIP Some students use the phrase, "Please Excuse My Dear Aunt
Sally," or the word "PEMDAS" (Parentheses, Exponents, Multiplication,
Division, Addition, Subtraction) to remind them of the order of operations. Re-
member: Multiplication and division are of the same order, and addition and sub-
traction are of the same order.

EXAMPLE 1 Evaluating an Expression

Evaluate the expression - x2 + 3x + 20 for x = 4.

SOLUTION: Substitute 4 for each x and use the order of operations to evaluate the
expression.

-x2 + 3x + 20
_(4)2 + 3(4) + 20
-16 + 12 + 20
-4 + 20
16

hiS painting by Charles Demuth, rEXAMPLE 2 Finding the Height

T called I Saw the Figure 5 in Gold,
depicts the abstract nature of num-
A ball is thrown upward off a bridge 40 feet above ground. Its height, in feet,
above ground, t seconds after it is thrown, can be determined by the expression
bers. Mathematician and philosopher
-16t2 + 30t + 40. Find the height of the ball, above ground, 2 seconds after it is
Bertrand Russell observed in 1919
that it must have required "many ages thrown.
to discover that a brace of pheasants SOLUTION: Substitute 2 for each t.
and a couple of days were both in-
stances of the number 2." The discov- -16t2 + 30t + 40
ery that numbers could be used not -16(2)2 + 30(2) + 40
only to count objects such as the num-
ber of birds but also to represent ab- -16(4) + 30(2) + 40
stract quantities represented a break- -64 + 60 + 40
through in the development of
-4 ± 40
algebra.
= 36

EXAMPLE 3 Substituting for Two Variables

2
Evaluate -3x + 2xy - 2i when x = 2 and y = 3.

r
SOLU~ION: Substitute 2 for each x and 3 for each y; then evaluate using the order of
operatIOns.

-3x2 + 2xy - 2i
= -3(2)2 + 2(2)(3) - 2(3f
= -3(4) + 2(2)(3) - 2(9)
= -12 + 12 - 18
= 0 - 18
= -18 ..•.
 •.•EXAMPLE 4 Is 3 a Solution?
Determine whether 3 is a solution to the equation 2x2 + 4x - 9 = 2l.

SOLUTION: To determine whether 3 is a solution to the equation, substitute 3 for

each x in the equation. Then evaluate the left-hand side of the equation using the or-
der of operations. If this leads to a 21 on the left-hand side of the equal sign, then
both sides of the equation have the same value, and 3 is a solution. In checking the
solution, we use ~ which means we are not sure if the statement is true.

2x2 + 4x - 9 21 1-

2(3)2 + 4(3) - 9 J:. 21

2(9) + 12 - 9 1- 21
18 + 12 - 9 1- 21
30 - 9 1- 21
21 = 21 True

Because 3 makes the equation a true statement, 3 is a solution to the equation. A

'-

Concept/Writing Exercises 20. 5x2 + 7x - II, x = -1

1. What is a variable? 21. ~x2 - 5x + 2, x = ~
2. What is a constant? 22. ~x2 + X - I, x = ~
3. What does it mean when we state that a number is a solu- 23. 8x3- 4x2
+ 7, x = ~
tion to an equation? 2
24. - x + 4xy, x = 2, y = 3
4. What is an algebraic expression? Illustrate an algebraic 25. 2x2 + xy + 3i, x = -2, y = 1
expression with an example.
26.3x2+~xy-ki, x=2, y=5
S. a) For the term 45, identify the base and the exponent.
b) In your own words, explain how to evaluate 45. 27. 4x2 - 12xy + 9i, x = 3, y = 2
28.(x+3y)2, x=4, y=-3
6. In your own words, explain the order of operations.
7. Evaluate 8 + 16 -;- 4 using the order of operations.
8. Evaluate 9 + 6· 3 using the order of operations. In Exercises 29-38, determine whether the valuers) is (are)
a solution to the equation.
29. 7x +3 = 23, x = 3
In Exercises 9-28, evaluate the expression for the given 30. 5x - 7 = -27, x = -4
valuer s) of the variable( s). 31. x - 3y = 0, x = 6, y = 3
9. x2, X = 7 10. x2, X = -8 32. 4x + 2y = -2, x = -2, y = 3
11. -x2, X = -3 12. - x2, x = - 5 33. x2 + 3x - 4 = 5, x = 2
13. -2x3, x = -7 14. -x3, x = -4 34. 2x2 - x - 5 = 0, x = 3
15. x -7, x = 4
16. 8x - 3, x = ~ 35. 2x2 + x = 28, x = -4
17. -7x + 4, x = -2 36.y=x2+3x-5, x=l, y=-l
18. x2 - 3x + 8, x = 5 2
37.y=-x +3x-1, x=3, y=-I
19. -x2 + 5x - 13, x = -2 38. y = x3 - 3x2 + 1, x = 2, y = - 3
 be determined by the expression 0.000002n. Determine the
number of seconds needed for the computer to do 8 trillion
39. Sales Tax If the sales tax on an item is 7%, the sales tax, in (8,000,000,000,000) calculations.
dollars, on an item costing d dollars can be found by using
44. Drying Time The time, in minutes, needed for clothes
the expression 0.07d. Determine the sales tax on a tele-
hanging on a line outdoors to dry, at a specific temperature
scope costing $175.
and wind speed, depends on the humidity, h. The time can
be approximated by the expression 2h2 + 80h + 40,
where h is the percent humidity expressed as a decimal
number. Find the length of time required for clothing to
dry if there is 60% humidity.
45. Grass Growth The rate of growth of grass in inches per
week depends on a number of factors, including rainfall
and temperature. For a certain area, this can be approxi-
mated by the expression 0.2R2 + 0.003RT + 0.0001 T2,
where R is the weekly rainfall, in inches, and T is the aver-
age weekly temperature, in degrees Fahrenheit. Find the
amount of growth of grass for a week in which the rainfall
is 2 in. and the average temperature is 70°F.

Challenge Problems/Group Activities

40. Radius of a Circle A pebble is dropped into a calm pond
causing ripples in the form of concentric circles (one circle
46. Explain why ( - J)" = I for any even number n.
inside another circle). The radius of the outer circle, in 47. Does (x + y)2 = x2 + y2? Complete the table and state
feet, can be determined by the expression 0.5/, where I is your conclusion.
the time in seconds after the pebble strikes the water. Find
x y (x + y)2 x2 + y2
the radius of the outer circle 3 sec after the pebble strikes
the water.
2 3
41. Cost of a Tour The cost, in dollars, for Crescent City -2 -3
Tours to provide a tour for x people can be determined by -2 3
the expression 220 + 2.75x. Determine the cost for Cres- 2 -3
cent City Tours to provide a tour for 75 people.
42. Orange Orchard The number of baskets of oranges that 48. Suppose 11 represents any natural number. Explain why I"
are produced by x trees in a small orchard can be approxi- equals I?
mated by the expression 25x - 0.2x2 (assuming x is no
more than 100). Find the number of baskets of oranges Internet/Research Activity
produced by 60 trees. 49. When were exponents first used? Write a paper explaining
43. 8 Trillion Calculations If a computer can do a calculation how exponents were first used and when mathematicians
in 0.000002 sec, the time required to do n calculations can began writing them in the present form.

6.2 LINEAR EQ1JATIONS IN ONE VARIABLE

In Section 6.1, we stated that two algebraic expressions joined by an equal sign form an
equation. The solution to some equations, such as x + 3 = 4, can be found easily by
trial and error. However, solving more complex equations, such as 2x - 3 = 4(x + 3),
requires understanding the meaning of like terms and learning four basic properties.
The parts that are added or subtracted in an algebraic expression are called terms.
The expression 4x - 3y - 5 contains three terms, namely 4x, -3y, and -5. The +
and - signs that break the expression into terms are a part of the terms. When listing
the terms of an expression, however, it is not necessary to include the + sign at the
beginning of the term.
 The numerical part of a term is called its numerical coefficient or, simply, its
coefficient. In the term 4x, the 4 is the numerical coefficient. In the term -4y, the -4
is the numerical coefficient.
Like terms are terms that have the same variables with the same exponents on the
variables. Unlike terms have different variables or different exponents on the
variables.

2x, 7x (same variable, x) 2x, 9 (only first term has a variable)

-8y, 3y (same variable, y) Sx, 6y (different variables)
-4, 10 (both constants) x, 8 (only first term has a variable)
-Sx2, 6x2 (same variable with same exponent) 2x3, 3x2 (different exponents)

To simplify an expression means to combine like terms by using the commuta-

tive, associative, and distributive properties discussed in Chapter S. For convenience,
we list these properties below.

Properties of the Real Numbers

a(b + c) = ab + ac Distributive property
a+b=b+a Commutative property of addition
ab = ba Commutative property of multiplication
(a + b) + c = a + (b + c) Associative property of addition
(ab)c = a(bc) Associative property of multiplication

Combine like terms in each expression.

a) 7x + 3x b) 6y - 2y

SOLUTION:
a) We use the distributive property (in reverse) to combine like terms.
I

7x + 3x = (7 + 3)x
lOx

b) 6y - 2y = (6 - 2)y = 4y
I c) x + 12 - 3x + 7 = x - 3x + 12 + 7 Rearrange terms, place like terms together.

-2x + 19 Combine like terms.

d) -2x + 71- - 6y - LI - Sy + 3x
= -2x + 3x - 6y - Sy + 4 - 11
 We are able to rearrange the terms of an expression, as was done in Example 1(c)
DID YOU KNOW and (d) by the commutative and associative properties that were discussed in Section
5.5.
Th,e order of the terms in an expression is not crucial. However, when listing the
terms of an expression we generally list the terms in alphabetical order with the con-

G reek philosopher Diophantus of ,

Alexandria (A.D. 250), who in- I
vented notations for powers of a
stant, the term without a variable, last.

number and for multiplication and Solving Equations

division of simple quantities, is
Recall that to solve an equation means to find the value or values for the variable that
thought to have made the first at-
tempts at algebra. But not until the make(s) the equation true. In this section, we discuss solving linear (or first degree)
sixteenth century did French mathe- equations. A linear equation in one variable is one in which the exponent on the vari-
matician Fran'rois Viete (1540-1603) able is 1. Examples of linear equations are 5x - 1 = 3 and 2x + 4 = 6x - 5.
use symbols to represent numbers, Equivalent equations are equations that have the same solution. The equations
the foundation of symbolic algebra. 2x - 5 = 1,2x = 6, and x = 3 are all equivalent equations since they all have the
However, the work of Rene Descartes same solution, 3. When we solve an equation, we write the given equation as a series
(1596-1660) is considered to be the of simpler equivalent equations until we obtain an equation of the form x = c, where
starting point of modern-day algebra. c is some real number.
In 1707, Sir Isaac Newton (1643-
To solve any equation, we have to isolate the variable. That means getting the
1727) gave symbolic mathematics
variable by itself on one side of the equal sign. The four properties of equality that we
the name universal arithmetic.
are about to discuss are used to isolate the variable. The first is the addition property.

Addition Property of Equality

If a = b, then a + c = b + c for all real numbers a, b, and c.

The addition property of equality indicates that the same number can be added to
both sides of an equation without changing the solution.

rEXAMPLE 2 Using the Addition Property of Equality

Find the solution to the equation x - 7 = 10.

SOLUTION: To isolate the variable, add 7 to both sides of the equation.

x-7=10
x-7+7=1O+7
x+O=17
x = 17

x-7llO
17 - 7 l 10 Substitute 17 for x.

10 = 10 True

In Example 2, we showed the step x +0 = 17. Generally this step is done men-
tally, and the step is not listed.

Subtraction Property of Equality

If a = b, then a - c = b - c for all real numbers a, b, and c.
 The subtraction property of equality indicates that the same number can be sub-
tracted from both sides of an equation without changing the solution.

EXAMPLE 3 Using the Subtraction Property of Equality

I Find the solution to the equation x +7 = 15.

SOLUTION: To isolate the variable, subtract 7 from both sides of the equation.

x +7 = 15
x +7 - 7 = 15 - 7
x = 8

Notethat we did not subtract 15 from both sides of the equation, since this would
l not result in getting x on one side of the equal sign by itself. .•.

Multiplication Property of Equality

If a = b, then a . C = b· c for all real numbers a, b, and c, where c =I- O.

The multiplication property of equality indicates that both sides of the equation
can be multiplied by the same nonzero number without changing the solution.

EXAMPLE 4 ~Sing the Multiplication Property of Equality

r Find the solution to "6 = 3.

SOLUTION: To solve this equation, multiply both sides of the equation by 6.

x
-=3
6

6(:~) 6(3) =

1
6x
18
6
1

Ix = 18
x= 18

6x
In Example 4, we showed the steps - = 18 and Ix = 18. Generally, we will not
illustrate these steps. 6

Division Property of Equality

a b
If a = b, then - = - for all real numbers a, b, and c, c =I- O.
c c
 The division property of equality indicates that both sides of an equation can be
divided by the same nonzero number without changing the solution. Note that the di-
visor, c, cannot be 0 because division by 0 is not permitted.

EXAMPLE 5 Using the Division Property of Equality

I Find the solution to the equation 3x = 15.

SOLUTION: To solve this equation, divide both sides of the equation by 3.

3x = 15
3x 15
-
3 3
x = 5 •••
An algorithm is a general procedure for accomplishing a task. The following
general procedure is an algorithm for solving linear (or first-degree) equations. Some-
times the solution to an equation may be found more easily by using a variation of this
general procedure. Remember that the primary objective in solving any equation is to
isolate the variable.

A General Procedure for Solving Linear Equations

1. If the equation contains fractions, multiply both sides of the equation by the low-
est common denominator (or least common multiple). This step will eliminate all
fractions from the equation.
2. Use the distributive property to remove parentheses when necessary.
3. Combine like terms on the same side of the equal sign when possible.
4. Use the addition or subtraction property to collect all terms with a variable on one
side of the equal sign and all constants on the other side of the equal sign. It may
be necessary to use the addition or subtraction property more than once. Thjs
process will eventually result in an equation of the form ax = b, where a and b
are real numbers.
S. Solve for the variable using the division or multiplication property. Thjs will result
in an answer in the form x = c, where C is a real number.

-EXAMPLE 6 Using the General Procedure

Solve the equation 2x - 9 = 19 and check your solution.

SOLUTION: Our goal is to isolate the variable; therefore, we start by getting the
term 2x by itself on one side of the equation.
2x - 9 = 19
2x - 9 + 9 = 19 + 9
2x = 28
2x 28
2 2
x = 14
 r EXAMPLE 7 Solving a Linear Equation
DID YOU KNOW
Solve the equation 4 = 5 + 2(t + I) for f.

A New- Corteept SOLUTION: Our goal is to isolate the variable t. To do so, follow the general proce-
dure for solving equations.

4 = 5 + 2(t + 1)
4 = 5 + 2t + 2 Distributive property (step 2)

4 = 2t + 7 Combine like terms (step 3)

4 - 7 = 2f + 7 - 7 Subtraction property (step 4)

-3 = 2t
3 2t
-- -
2 2
3
-- = t
2

W ell into the sixteenth century,

mathematicians found it diffi-
cult to accept the idea that the solu- EXAMPLE 8 Solving an Equation Containing Fractions
tion to a problem (such as Example
7) could be a negative number be-
cause negative numbers could not be
I Solve the equation
2x
3 +
1
3" = "4'
3

accepted as physically real. In the

early days of algebra, someone
working a problem did not isolate a
I SOLUTION: When an equation contains fractions, we generally begin by multiply-
ing each term of the equation by the lowest common denominator, LCD (see
Chapter 5). In this example, the LCD is 12, since 12 is the smallest number that is
variable by subtracting like terms.
Instead, a problem would be put into I divisible by both 3 and 4.
a form that allowed only positive co-
efficients and answers. Albert Girard
(l595-J637) contributed to the evo- 12C; +~)
lution of a correct understanding of
negative quantities. 12( 2;) + 12( ~ ) 12( ~ ) Distributive property (step 2)

~( ~) + ~(~) = h( ~) Divide out common factors.

I I I
8x + 4 = 9
8x +4 - 4 =9 - 4 Subtraction property (step 4)

8x = 5
8x 5
8 8
5
x=-
8

A check will show that ~ is the solution to the equation. You could have worked the
problem without first multiplying both sides of the equation by the LCD. Try it!

r EXAM P LE 9 Variables on Both Sides of the Equation

Solve the equation 6x + 8 = lOx + 12.
 SOLUTION: Note that the equation has an x on both sides of the equal sign. In equa-
tions of this type, you might wonder what to do first. It really does not matter as
long as you do not forget the goal of isolating the variable x. Let's collect the terms
containing a variable on the left-hand side of the equation.

6x + 8 = lOx + 12
6x + 8- 8 = lOx + 12 - 8 Subtraction property (step 4)

6x = lOx + 4
6x - lOx = lOx - lOx + 4 Subtraction property (step 4)

-4x = 4

-4x 4
-4 -4

In the solution to Example 9, the terms containing the variable were collected on
the left-hand side of the equal sign. Now work Example 9, collecting the terms with
the variable on the right-hand side of the equal sign. If you do so correctly, you will
get the same result.

EXAMPLE 10 Solving an Equation Containing Decimals

Solve the equation 4x - 0.48 = 0.8x + 4 and check your solution.

SOLUTION: This problem may be solved with the decimals, or you may multiply
each term by 100 and eliminate the decimals. We will solve the problem with the
decimals.

4x - 0.48 = 0.8x +4
4x - 0.48 + 0.48 = 0.8x + 4 + 0.48
4x = 0.8x + 4.48
4x - 0.8x = 0.8x - 0.8x + 4.48
3.2x = 4.48
3.2x 4.48
-- --
3.2 3.2
x = 1.4

4x - 0.48 = 0.8x +4
4( 1.4) - 0.48 = 0.8( 1.4) +4 Substitute 1.4for each x in the equation.

5.6 - 0.48 = 1.12 +4

5.12 = 5.12 True A

In Chapter 5, we explained that a - b can be expressed as a + (-b). We use

this principle in Example 11.

rEXAMPLE 11
Solve 10 = -5
Using the Definition

+ 3(p - 4) for p.
of Subtraction
 SOLUTION: Our goal is to isolate the variable p. To do so, follow the general proce-
dure for solving equations.
10 = -5 + 3(p - 4)
10 = -5 + 3[p + (-4)]
10 = -5 + 3(p) + 3(-4)
10 = -5 + 3p - 12
10 = 3p - 17
10 + 17 = 3p - 17 + 17 Addition property

27 = 3p Combine like terms.

27 3p
- -
3 3
9=p

TIMELY TIP Remember that the goal in solving an equation is to get the vari-
able alone on one side of the equal sign.

So far, every equation has had exactly one solution. Some equations, however,
have no solution and others have more than one solution. Example 12 illustrates an
equation that has no solution, and Example 13 illustrates an equation that has an infi-
nite number of solutions.

I EXAMPLE 12 An Equation with No Solution

Solve 3(x - 4) +x +2 = 6x - 2(x + 3).

SOLUTION:

3(x - 4) + x + 2 = 6x - 2(x + 3)
3x - 12 + x + 2 = 6x - 2x - 6 Distributive property

4x - 10 = 4x - 6 Combine like terms.

4x - 4x - 10 = 4x - 4x - 6 Subtraction property

- 10 = -6 False

During the process of solving an equation, if you obtain a false statement like
-10 = -6, or -4 = 0, the equation has no solution. An equation that has no solu-
tion is called an inconsistent equation. The equation 3 (x - 4) + x + 2 =
6x - 2( x + 3) is inconsistent and thus has no solution.

, EXAMPLE 13 An Equation with Infinitely Many Solutions

Solve 3(x + 2) - 5(x - 3) = -2x + 21.
SOLUTION:

3 (x + 2) - 5 (x - 3) - 2x+ 21
3x + 6 - 5x + 15 = -2x + 21 Distributive property

-2x + 21 = -2x + 21 Combine like terms.

I Note that at this point both sides of the equation are the same. Every real number will
satisfy this equation. This equation has an infinite number of solutions. An equation
of this type is called an identity. When solving an equation, if you notice that the
same expression appears on both sides of the equal sign, the equation is an identity.
The solution to any linear equation that is an identity is all real numbers. If you con-
tinue to solve an equation that is an identity, you will end up with 0 = 0, as follows.

+ 21
-2x -2x + 21
-2x + 2x + 21 -2x + 2x + 21 Addition property

21 = 21 Combine like terms.

21 - 21 = 21 - 21 Subtraction property

o = 0 True for any value of x

Proportions
A ratio is a quotient of two quantities. An example is the ratio of 2 to 5, which can be
written 2 : 5 or ~ or 2/5.

e
An example of a proportion is t;a = d. Consider the proportion

x+2 x +5
5 8

We can solve this proportion by first multiplying both sides of the equation by the
least common denominator, 40.

x+2 x+5
5 8

A~(X :
'"fV 0 2) = }A"(X :
'"fV 0
5) Multiplication property

8(x+ 2) = 5(x + 5)
8x + 16 = 5x + 25
3x + 16 = 25
3x = 9
X = 3

A check will show that 3 is the solution.

Proportions can often be solved more easily by using cross multiplication.

Cross Multiplication
a e
Ift; = d' then ad = be,
 5 8
8 (x + 2) = 5 (x + 5) Cross multiplication

8x + 16 = 5x + 25
3x + 16 = 25
3x =9
X = 3

To Solve Application Problems Using Proportions

1. Represent the unknown quantity by a variable.
2. Set up the proportion by listing the given ratio on the left-hand side of the equal
sign and the unknown and other given quantity on the right-hand side of the equal
sign. When setting up the right-hand side of the proportion, the same respective
quantities should occupy the same respective positions on the left and right. For
example, an acceptable proportion might be
miles miles
-- --
hour hour
3. Once the proportion is properly written, drop the units and use cross multiplica-
tion to solve the equation.
4. Answer the question or questions asked.

rEXAMPLE 14 Water Usage

The cost for water in Orange County is $1.42 per 750 gallons (gal) of water used.
What is the water bill if 30,000 gallons are used?
I SOLUTION: This problem may be solved by setting up a proportion. One proportion
that can be used is

cost of 750 gal cost of 30,000 gal

750 gal 30,000 gal

We want to find the cost for 30,000 gallons of water, so we will call this quantity x.
I The proportion then becomes
1.42 X
Given ratio { 750 = 30,000

(1.42)(30,000) = 750x
42,600 = 750x
42,600 750x
--- --
750 750
$56.80 = x
 Insulin comes in 10 cubic centimeter (cc) vials labeled in the number of units of in-
sulin per cubic centimeter of fluid. A vial of insulin marked U40 has 40 units of in-
sulin per cubic centimeter of fluid. If a patient needs 30 units of insulin, how much
I fluid should be drawn into the syringe from the U40 vial?

SOLUTION: The unknown quantity, x, is the number of cubic centimeters of fluid to

be drawn into the syringe. Following is one proportion that can be used to find that
quantity.

40 units __ 30 units
Given ratio { I cc xcc
40x = 30(1)
40x = 30
30
X = - = 0.75
40

The nurse or doctor putting the insulin in the syringe should draw 0.75 cc of the
fluid.

Concept/Writing Exercises 21. -3x + 2 - 5x 22. - 3x + 4x - 2 + 5

1. Define and give an example of a term.
23. 2 - + I
3x - 2x 24. -0.2x + 1.7 x - 4
2. Define and give an example of like terms.
25. 6.2x - 8.3 + 7.1x 26. ~x + ix - 5

3. Define and give an example of a numerical coefficient.

27. ~x - kx - 4

4. Define and give an example of a linear equation.

28. 7t + 5s + 9 - 3t - 2s - 12
S. Explain how to simplify an expression. Give an example.
29. 5x - 4y - 3y + 8x + 3
6. State the addition property of equality. Give an example.
30. 3(p + 2) - 4(p + 3)
7. State the subtraction property of equality. Give an
31. 2(s + 3) + 6(s - 4) + I
example. 32. 6(r - 3) - 2(1' + 5) + 10
8. State the multiplication property of equality. Give an 33. 0.3(x + 2) + 1.2(x - 4)
example. 34. ~(x + 2) - -,'ox
9. State the division property of equality. Give an example. 35. ~x + ~ - ±x
10. Define algorithm. 36. n - ~ + ~n - i
11. Define and give an example of a ratio. 37. 0.5(2.6x - 4) + 2.3(IAx - 5)
12. Define and give an example of a proportion. 38.~(3x + 9) - ±(2x + 5)
13. Are 3x and ~ x like terms? Explain.
14. Are 4x and 4y like terms? Explain.
39. y + 8 = 13 40. 2y - 7 = 17

41.9= 12 - 3x 42. 14 = 3x +5
3 7 x-I x+5
43. - = - 44. --=--
15. 2x + 9x 16. -4x - 7x x 8 5 15
17. 5x - 3x + 12 18.-6x+3x+21 45. ~x + k = ~ 46. ~y +k=±
19.7x + 3y - 4x + 8y 20. x - 4x + 3 48. 5x + 0.050 = -0.732
 x 1
49. 6t - 8 = 4t - 2 50. - + 2x = -
4 3
x-3 x+4 x-5 x-9
51. -- = -- 52.
2 3 4 3
53. 6t - 7 = 8t + 9 54. 12x - 1.2 = 3x + 1.5
55. 2(x + 3) - 4 = 2(x - 4)
56. 3(x + 2) + 2(x - 1) = 5x - 7 a) How many kilometers per hour are equal to I mph?
b) On a stretch of the Queen Elizabeth Way, the speed
57. 4( x - 4) + 12 = 4( x-I)
limit is 90 kph. What is the speed limit in miles
y 2y per hour?
58. - +4 = - - 6
3 5 72. The Proper Dosage A doctor asks a nurse to give a patient
59.±(x + 4) = ~(x + 2) 250 milligrams (mg) of the drug Simethicone. The drug is
60.~(x + 5) = ±(x + 2) available only in a solution whose concentration is 40 mg
Simethicone per 0.6 millimeter (mm) of solution. How
61. 3x + 2 - 6x = - x - 15 + 8 - 5x
many millimeters of solution should the nurse give the
62.6x + 8 - 22x = 28 + l4x - 10 + 12x patient?
63. 2(t - 3) + 2 = 2(2t - 6)
64. 5.7x - 3.1(x + 5) = 7.3 Amount of Insulin In Exercises 73 and 74, how much in-
sulin (in cc) would be givenfor thefollowing doses? (Refer
Problem Solving to Example 15 on page 302.)
In Exercises 65 and 66, use the DeKalb County water rate 73. 12 units of insulin from a vial marked U40
of$2.05 per 1000 gallons of water used. 74. 35 units of insulin from a vial marked U40

65. Water Bill What is the water bill if a resident uses 35,300 75. a) In your own words, summarize the procedure to use to
gal? solve an equation.
b) Solve the equation 2(x + 3) = 4x + 3 - 5x with
66. Limiting the Cost How many gallons of water can the cus-
the procedure you outlined in part (a).
tomer use if the water bill is not to exceed $40.68?
76. a) What is an identity?
67. Dial Bodywash A bottle of Dial Bodywash contains 354 b) When solving an equation, how will you know if the
milliliters (m!) of soap. If Tony Vaszquez uses 6 ml for equation is an identity?
each shower, how many times can he shower using one 77. a) What is an inconsistent equation?
bottle of Dial Bodywash? b) When solving an equation, how will you know if the
68. Fajitas A recipe for six servings of beef fajitas requires 16 equation is inconsistent?
oz of beef sirloin.
a) If the recipe were to be made for nine servings, how Challenge Problems/Group Activities
many ounces of beef sirloin would be needed? 78. Depth of a Submarine The pressure, P, in pounds per square
b) How many servings of beeffajitas can be made with 32 inch (psi), exerted on an object x ft below the sea is given by
oz of beef sirloin? the formula P = 14.70 + 0.43x. The 14.70 represents the
69. Watching Television Nielson Media Research determines weight in pounds of the column of air (from sea level to the
the number of people who watch a television show. One rat- top of the atmosphere) standing over a I in. by I in. square of
ing point means that about 1,022,000 households watched seawater. The 0.43x represents the weight in pounds of a col-
the show. The top-rated television show for the week of umn of water I in. by I in. by x ft (see Fig. 6.1).
September 23,2002, was Friends, with a rating of 20.3.
About how many households watched Friends that week?
This column
70. Grass Seed Coverage A 20 Ib bag of grass seed will cover of air weighs
14.71b
an area of 10,000 ft2.
a) How many pounds are needed to cover an area of
140,000 ft2?
b) How many bags of grass seed must be purchased to
cover an area of 140,000 ft2?
I x ft
This column
of water
weighs 0.43x Ib
71. Speed Limit When Jacob Abbott crossed over from Nia-
gara Falls, New York, to Niagara Falls, Canada, he saw a 1
sign that said 50 miles per hour (mph) is equal to 80 kilo-
meters per hour (kph).
 a) A submarine is built to withstand a pressure of 148 psi.
How deep can that submarine go?
b) If the pressure gauge in the submarine registers a pres- 80. Ratio and proportion are used in many different ways in
sure of 128.65 psi, how deep is the submarine? everyday life. Submit two articles from newspapers, maga-
zines, or the Internet in which ratios and/or proportions are
79. a) Gender Ratios If the ratio of males to females in a used. Write a brief summary of each article explaining
class is 2 : 3, what is the ratio of males to all the stu- how ratio and/or proportion were used.
dents in the class? Explain your answer. 81. Write a report explaining how the ancient Egyptians used
b) If the ratio of males to females in a class is m : n, what equations. Include in your discussion the forms of the
is the ratio of males to all the students in the class? equations used.

A formula is an equation that typically has a real-life application. To evaluate a for-

mula, substitute the given values for their respective variables and then evaluate using
the order of operations given in Section 6.1. Many of the formulas given in this sec-
tion are discussed in greater detail in other parts of the book.

EXAMPLE 1 Simple Interest

r The simple interest formula*, interest = principal X rate X time, or i = prt, is

used to find the interest you must pay on a simple interest loan when you borrow

I principal, p, at simple interest rate, r, in decimal form, for time, t. Chris Campbell
borrows $3000 at a simple interest rate of 9% for 3 years.

I a) How much will Chris Campbell pay in interest at the end of 3 years?
b) What is the total amount he will repay the bank at the end of 3 years?

SOLUTION:

I a) Substitute the values of p, r, and t into the formula; then evaluate.

i = prt
= 3000(0.09)(3)
=810

Thus, Chris must pay $810 interest.

b)The total he must pay at the end of 3 years is the principal, $3000, plus the $810

l interest, for a total of $3810.

EXAMPLE 2 Volume of a Cereal Box

r The
V
formula for the volume of a box* is volume
lwh. Use the formula V
=
length X width X height or
=
lwh to find the width of a Sweet Treats cereal box
=
I if I = 7.5 in., h = 10.5 in., and V = 196.875 in.3.
 SOLUTION: We substitute the appropriate values into the volume formula and solve
for the desired quantity, w.
v = lwh
196.875 = (7.5)w(IO.5)
196.875 = 78.75w
196.875
=w
78.75
2.5 = w

In Example 1, we used the formula i = prt. In Example 2, we used the formula

V = lwh. In these examples, we used a mathematical equation to represent real phe-
nomena. When we represent real phenomena, such as finding simple interest, mathe-
matically we say we have created a mathematical model or simply a model to repre-
sent the situation. A model may be a single formula, or equation, or a system of many
equations. By using models we gain insight into real-life situations, such as how
much interest you will accumulate in your savings account. We will use mathematical
models throughout this chapter and elsewhere in the text. In some exercises in this
and the next chapter, when you are asked to determine an equation to represent a real-
life situation, we will sometimes write the word model in the instructions.
Many formulas contain Greek letters, such as JL (mu), (J" (sigma), L (capital
sigma), {)(delta), E (epsilon), 7T (pi), (J (theta), and A (lambda). Example 3 makes use
of Greek letters.

X-JL
z=---
(J"

vn

X-JL 120 - 100 20 20

z=---= =-=-=25
(J" 16 16 8 .
vn V4 2

Some formulas contain subscripts. Subscripts are numbers (or letters) placed be-
low and to the right of variables. They are used to help clarify a formula. For example,
if two different amounts are used in a problem, they may be symbolized as A and Ao,
or Al and A2. Subscripts are read using the word sub; for example, Ao is read "A sub
zero" and Al is read "A sub one."
 Exponential Equations
Many real-life problems, including population growth, growth of bacteria, and decay
Global E-commerce Revenue
of radioactive substances, increase or decrease at a very rapid rate. For example, in
In billions
Fig. 6.2, which shows global electronic business revenue, in billions of dollars, from
1996 through 2003, the graph is increasing rapidly. This is an example where the
graph is increasing exponentially. The equation of a graph that increases or decreases
exponentially is called an exponential equation (or exponential formula). An expo-
nential equation is of the form y = aX, a > 0, a *
1. We often use exponential equa-
tions to model real-life problems. In Section 6.10, we will discuss exponential equa-
tions (and exponential formulas) in more detail.
In an exponential formula, letters other than x and y may be used to represent the
variables. The following equations are examples of exponential formulas:
y = 2x, A = (~y, and P = 2.31• Note in the exponential formula that the variable is
the exponent of some positive constant that is not equal to I. In many real-life appli-
cations, the variable t will be used to represent time. Problems involving exponential
formulas can be evaluated much more easily if you use a calculator contajning a
[2],0,or5] key.
The following formula, referred to as the exponential growth or decay formula,
is used to solve many real-Ijfe problems.

In the formula, Po represents the original amount present, P represents the amount
present after t years, and a and k are constants.
When k > 0, P increases as t increases and we have exponential growth. When
k < 0, P decreases as t increases and we have exponential decay.

EXAMPLE 4 Using an Exponential Decay Formula

Carbon datjng is used by scientists to find the age of fossils, bones, and other items.
[
The formula used in carbon dating is

where Po represents the orjginal amount of carbon 14 (CI4) present and P repre-
sents the amount of C 14 present after t years. If 10 mg of C 14 is present in an animal
bone recently excavated, how many milligrams will be present in 3000 years?

SOLUTION: Substituting the values in the formula gives

P = poTt/5600
P = 10(2)-3000/5600

P ::::::10(2)-0.54 (Recall that;:::: means "is approximately equal to")

P ::::::10(0.69)
p:::::: 6.9 mg

Thus, in 3000 years, approximately 6.9 mg of the original 10 mg of CI4 will re-
main.
 In Example 4, we used a calculator to evaluate (2f3000/5600. The steps used to
find this quantity on a calculator with a [2]
key are

2 [2] 0] 3000 ~ G 5600 OJB .6898170602

After the B key is pressed, the calculator displays the answer 0.689817. To evalu-
ate 10(2)-3000/5600 on a scientific calculator, we can press the following keys.

10 02 [2]0] 3000 ~G 5600 OJB 6.898170602

Notice that the answer obtained using the calculator steps shown above is a little more
accurate than the answer we gave when we rounded the values before the final answer
in Example 4.
When the a in the formula P = Poakt is replaced with the very special letter e, we
get the naturaL exponentiaL formuLa

The letter e represents an irrational number whose value is approximately 2.7183. The
number e plays an important role in mathematics and is used in finding the solution to
many application problems.
To evaluate e(0.04)5 on a calculator, as will be needed in Example 5, press*

0] .040 5 OJ~~ 1.221402758

7' '\
Inverse Natural
key logarithm
key

After the ~ key is pressed, the calculator displays the answer 1.221402758.
To evaluate 1O,000e(0.04)5 on a calculator, press

10000 ~1[I1.04 ~15 [[1~1~IEI12214.02758

Inthis calculation, after the B key is pressed, the calculator displays the answer
12214.02758.

EXAMPLE 5 Using an ExponentiaL Growth FormuLa

r Banks often credit compound interest continuously. When that is done, the principal
amount in the account, P, at any time t can be calculated by the natural exponential
formula P = Poekt, where Po is the initial principal invested, k is the interest rate in
decimal form, and f is the time.
 Suppose $10,000 is invested in a savings account at a 4% interest rate com-

I
pounded continuously. What will be the balance (or principal) in the account in
5 years?
SOLUTION:

P = Poekt
10,000e (004)5
1O,000e (0.20)
~ 10,000(1.221402758)
~ 12,214

Thus, after 5 years, the account's value will have grown from $10,000 to about
$12,214, an increase of about $2214.

Graphing calculators are a tool that can be used to graph equations. Figure 6.3
shows the graph of P = 1O,000eo.04t as it appears on the screen (or window) of a
Texas Instrument TI-83 Plus graphing calculator. To obtain this screen, the domain (or
the x-values) and range (or the y-values) of the window need to be set to selected val-
ues. We will speak a little more about graphing calculators shortly. In Section 6.10,
we will explain how to graph exponential equations by plotting points.

EXAMPLE 6 Population of Nevada

r The population of Nevada, which was the fastest growing state in every decade of
the twentieth century except for the 1950s, is continuing to grow exponentially at
the rate of about 5.10% per year. In 2000, the population of Nevada was 1,998,257.
Nevada's expected population, t years after 2000 is given by the formula
P = 1,998,257eo.05101. Find the expected population of Nevada in the year 2010.

SOLUTION: Since 2010 is 10 years after 2000, t = 10 years.

P = 1,998,257eo.051Ot
1,998,257eo.0510( 10) Substitute 10 for t.

= 1,998,257eo.510
~ 1,998,257(1.665291195)
~ 3,327,679.787

Thus, in the year 2010, the population of Nevada is expected to be about 3,327,697
people. .•.

TIMELY TIP When doing calculations on the calculator, do not round any value
before obtaining the final answer. By not rounding, you will obtain a more accu-
rate answer. For example, if we work Example 6 on a calculator, and rounded
eO.5IO to 1.67, we would determine that the population of Nevada in 2010 is ex-
pected to be about 3,337,089 which is a less accurate answer.
DID YOU KNOW Solving for a Variable in a Formula or Equation
Often in mathematics and science courses, you are given a formula or an equation ex-
pressed in terms of one variable and asked to express it in terms of a different vari-
able. For example, you may be given the formula P = Fr and asked to solve the for-
mula for r. To do so, treat each of the variables, except the one you are solving for, as
if it were a constant. Then solve for the variable desired, using the properties previ-
ously discussed. Examples 7 through 9 show how to do this task.
When graphing equations in Section 6.7, you will sometimes have to solve the
equation for the variable y as is done in Example 7.

EXAMPLE 7 Solving an Equation Containing More Than One Variable

Solve the equation 2x + 5y - 10 = 0 for y.

SOLUTION: We need to isolate the term containing the variable y. Begin by moving
the constant, -10, and the term 2x to the right-hand side of the equation.

M any of you recognize the for-

mula E = mc2 used by Albert
Einstein in 1912 to describe his
2x + 5y - 10 = 0
2x + 5y - 10 + 10 = 0 + 10
groundbreaking theory of relativity.
In the formula, E is energy, m is 2x + 5y = 10
mass, and c is the speed of light. In -2x + 2x + 5y = -2x + 10
his theory, Einstein hypothesized
5y = -2x + 10
that time was not absolute and that
mass and energy were related. 5y -2x + 10
Einstein's 72-page handwritten man- 5 5
uscript, which went on display in -2x + 10
1999 at Jerusalem's Israel Museum,
5
shows that Einstein toyed with using
an L rather than an E to represent en- 2x 10
y = -- +-
ergy. He wrote L = mc2, and then 5 5
scratched out the L and replaced it
2
with an E to produce the equation y = --x +2
E = mc2. Yisrail Galili at 5
Jerusalem's Hebrew University said
the L might have been a lambda, a Note that once you have found y = -2,; 5+ 10, you have solved the equation for y.
Greek letter sometimes used in func- The solution can also be expressed in the form y = -~ x + 2. This form of the equa-
tions related to energy. He said it tion is convenient for graphing equations, as will be explained in Section 6.7. Exam-
might also be related to Joseph La- ple 7 can also be solved by moving the y term to the right-hand side of the equal sign.
grange, an eighteenth-century Italian Do so now and note that you obtain the same answer.
mathematician.

x-p,
if
 SOLUTION: To isolate the term x, use the general procedure for solving linear equa-
tions given in Section 6.2. Treat each letter, except x, as if it were a constant.

x-J.-L
z=---
u

zu = x - J.-L

zu + J.-L = x - J.-L + J.-L Add JJ- to both sides of the equation.

ZU + J.-L = x
or x = Zu + J.-L A..

EXAMPLE 9 The Tax Free Yield Formula

r A formula that may be important to you now or sometime in the future is the tax-
free yield formula, Tf = Ta( 1 - F). This formula can be used to convert a taxable
yield, Ta, into its equivalent tax-free yield, Tf, where F is the federal income tax
bracket of the individual. A taxable yield is an interest rate for which income tax is
paid on the interest made. A tax-free yield is an interest rate for which income tax
does not have to be paid on the interest made.

PROFILE IN
MATHEMATICS

ecause she was a woman, Sophie Germain (1776-1831) was denied admission to the Ecole
B Polytechnic, the French academy of mathematics and science. Not to be stopped, she ob-
tained lecture notes from courses in which she had an interest, including one taught by Joseph-
Louis Lagrange. Under the pen name M. LeBlanc, she. submitted a paper on analysis to La-
grange, who was so impressed with the report that he wanted to meet the author and personally
congratulate "him." When he found out that the author was a woman, he became a great help and
encouragement to her. Lagrange introduced Germain to many of the French scientists of the time.
In 1801, Germain wrote the great German
mathematician Carl Friedrich Gauss to discuss
Fermat's equation, x" + y" = Z". He com-
mended her for showing "the noblest courage,
quite extraordinary talents and a superior ge-
nius." Germain's interests included work in
Germain was the first person to devise a formula
number theory and mathematical physics. She
describing elastic motion. The study of the equa- would have received an honorary doctorate
tions for the elasticity of different materials from the University of Gottingen, based on
aided the development of acoustical diaphragms Gauss's recommendation, but died before the
in loudspeakers and telephones. honorary doctorate could be awarded.
 a) For someone in a 25% tax bracket, find the equivalent tax-free yield of a 4%
taxable investment.
b) Solve this formula for Ta. That is, write a formula for taxable yield in terms of
tax-free yield.

SOLUTION:

a) Tf = Ta(l - F)
= 0.04( 1 - 0.25) = 0.04(0.75) = 0.03, or 3%

Thus, a taxable investment of 4% is equivalent to a tax-free investment of 3%

for a person in a 25% income tax bracket.

b) Tf = Ta(l - F)
Tf Ta{1--F}
1 - F J...---F
Tf Tf
1- F = Ta, or T =--
a 1- F

Concept/Writing Exercises 13. A = 7f(RZ - rZ); find A when R = 6,7f = 3.14, and
r = 4 (geometry).
1. What is a formula?
703w
2. Explain how to evaluate a formula. 14. B =
--z-; find B when w = 130 and h = 67 (for
h
3. What are subscripts? finding body mass index).
4. What is the simple interest formula? X-JL
5. What is an exponential equation? 15. z = --; find JL when z = 2.5, x = 42.1, and
tJ"
6. a) In an exponential equation of the form y = aX, what tJ" = 2 (statistics).
are the restrictions on a? 16. S = B + ~P s; find P when s = 10, S = 300, and
b) In an exponential equation of the form y = Poakt, B = 100 (geometry).
what does Po represent?
PV
17. T = -; find P when T = 80, V = 20, and k = 0.5
k
Practice the Skills (physics).
In Exercises 7-38, use the formula to find the value of the a+b+c
indicated variable for the values given. Use a calculator 18. m = ; find a when m = 70, b = 60, and
3
when one is needed. When necessary, round answers to the c = 90 (statistics).
nearest hundredth.
19. A = P( I + rt); find P when A = 3600, r = 0.04,
7. P = 4s; find P when s = 5 (geometry). and t = 5 (economics).
8. P = a + b + c; find P when a = 25, b = 53 and a +b
c = 32 (geometry). 20. m = --; find a when m = 70 and b = 77
2
9. P = 2l + 2w; find P when l = 12 and w = 16 (statistics).
(geometry). 21. v = ~atZ; find a when v = 576 and t = 12 (physics).
10. F = ma; find m when F = 40 and a = 5 (physics). 22. F = ~C + 32; find F when C = 7 (temperature
11. E = mcz; find m when E = 400 and c = 4 (physics). conversion).
12. p = iZr; find r when p = 62,500 and i = 5 23. C = ~(F - 32); find C when F = 77 (temperature
(electronics). conversion).
 F - 32
24. K = --- + 273.J; find K when F = 100
1.8
49. E = I R for R
(chemistry).
51. P = a + b + c for a
25. m = Y2 - YI; find m when Y2 = 8, YI = -4, 52. p = a + b + Sl + S2 for s,
x2 - xI
53. V = ~ Bh for B 54. V = 7Tr2h for h
x2 = -3, and x, = -5 (mathematics).
2gm
X-fL 56. r = -?- for m
26 .....
7 = --'
u' find z when:X = 66, fL = 60, U = 15, c-
58. y = mx +b
vn
60. A = d1d2
and n = 25 (statistics).
2
27. S = R - rR; find R when S = 186 and r = 0.07 a+b+c
(for determining sale price when an item is discounted). 61. A = ----
3
for c 62. A = k bh for b
28. S = C + rC; find Cwhen S = lIS and r = 0.15 KT P1VI P2V2
(for determining selling price when an item is marked up). 63. P = - forT 64. -- = -- for V2
V T, T2
29. E = alPI + a2P2 + a3P3; find E when
65. F = ~C + 32 for C 66. C = ~(F - 32) for F
al = 5, P, = 0.2, a2 = 7, P2 = 0.6, a3 = 10, and
P3 = 0.2 (probability). 67. S = 7Tr2 + 7Trs for s
-b + Vb2 - 4ac
30. X = 2a ; tind x when a = 2, b = -5,
and c = -12 (mathematics). Problem Solving
31. S = -16t2 + vot + so; find s when t = 4, Vo = 30, and 69. Refund Check Joel and Patti Karpel received a $600 in-
So = 150 (physics). come tax refund check from the federal government and
32. R = 0 + (V - D)r; find 0 when R = 670, decided to deposit the check in a money market account
V = 100, D = 10, and r = 4 (economjcs). that paid 2% simple interest per year. Determine
a) how much interest was added to their account at the end
33. P = _f_; findfwhen i = 0.08 and P = 3000 of 1 year.
1+ i
b) the balance in their account at the end of 1 year.
(investment banking).
70. Interest on a Loan Jeff Hubbard borrowed $800 from his
34. c = Va
2 + b2; find c when a = 5 and b = 12
brother for 2 years. At the end of 2 years, he repaid the $800
(geometry).
plus $128 in interest. What simple interest rate did he pay?
Gmlm2 71. VoLume in a Soup Can Determine the volume of a cylin-
35.F= 2; findGwhenF=625,mj 100,
r drical soup can if its diameter is 2.5 in. and its height is
m2 = 200, and r = 4 (physics). 3.75 in. (The formula for the volume of a cylinder is
nRT V = 7Tr2h. Use your pi key, EI, on your calculator, or
36.P=--;
V
findVifP= 12,n= lO,R=60,and
3.l4 for 7T if your calculator does not have a EI key.)
T = 8 (chemistry). Round your answer to the nearest tenth.
a, (1 - r")
37. SII =
I - r
; find 511 when a, = 8, r = t and
n = 3 (mathematics).

38. A = P ( I + -;;r)"l ; find A when P = 100, r = 6%,

n = I, and t = 3 (bankjng).

39. lOx - 4y = 13 40. 8x - 6 y = 2I

41. 4x + 7y = 14 42. -2x + 4y = 9
43. 2x - 3y + 6 = ° 44. 3x + 4y = ° 72. Body Mass Index A person's body mass index (BMI) is
45. -2x + 3y +z = IS 46. 5x + 3y - 2z = 22 703w
found by the formula B = 7' where w is the person's
47. 9x + 4z = 7 + 8y 48. 2x - 3y + 5z = °
 weight, in pounds, and h is the person's height, in inches. Challenge Problem/Group Activity
Lance Bass is 6 ft tall and weighs 200 lb.
77. Determine the volume of the block shown in Fig. 6.4, ex-
a) Determine his BMI.
cluding the hole.
b) If Lance would like to have a BMI of 26, how much
weight would he need to gain or lose?
73. Bacteria The number of a certain type of bacteria, y, present
in a culture is determined by the equation y = 2000(3 y,
where x is the number of days the culture has been growing.
Find the number of bacteria present after 5 days.
74. Adjustingfor Inflation If P is the price of an item today,
the price of the same item n years from today, P", is
P" = P( 1 + r )", where r is the constant rate of inflation.
Determine the price of a movie ticket 10 years from today
if the price today is $8.00 and the annual rate of inflation 78. Triangle Seek In this word seek, you are looking for six-
is constant at 3%. letter words that form triangles. The first letter of the words
75. Value of New York City Assume the value of the island of can be in any position in the triangle; the remaining letters
Manhattan has grown at an exponential rate of 8% per year of the words are in order moving clockwise or counterclock-
since 1626 when Peter Minuit of the Dutch West India wise. Triangles may overlap other triangles and the triangles
Company purchased the island for $24. The value of the can point up or down. For example, the word LINEAR is in-
island, V, at any time, t, in years after 1626, can be found dicated below. The word list is below on the right.
by the formula V = 24eo.08/. What is the value of the is-
S FAR R LID G R T P Word List
land in 2003, 377 years after Minuit purchased it? C C J 0 ANY E B 0 L F LINEAR
76. Radioactive Decay Strontium 90 is a radioactive isotope that R lED N END 0 M S 0 SYMBOL
decays exponentially at a rate of 2.8% per year. The amount TEE E W TOM E Y I H DEGREE
of strontium 90, S, remaining after t years can be found by F S ERG H T L Y Z M T
FACTOR
the formula S = Soe-O.028/, where So is the original amount A P J N R W C I N CUD
present. If there are originally 1000 g of strontium 90, find MOP PRO L R N A T A GROWTH
the amount of strontium 90 remaining after 30 years. G R 0 lYE U U FRO U NEWTON

6.4 APPLICATIONS OF LINEAR EQlJATIONS

IN ONE VARIABLE
One of the main reasons for studying algebra is that it can be used to solve everyday
problems. In this section, we will do two things: (I) show how to translate a written
problem into a mathematical equation and (2) show how linear equations can be used in
solving everyday problems. We begin by illustrating how English phrases can be written
as mathematical expressions. When writing a mathematical expression, we may use any
letter to represent the variable. In the following illustrations, we use the letter x.

Phrase Mathematical expression

Six more than a number x +6
A number increased by 3 x +3
Four less than a number x-4
A number decreased by 9 x-9
Twice a number 2x
Four times a number 4x
3 decreased by a number 3 - x
The difference between a number and 5 x - 5
 Sometimes the phrase that must be converted to a mathematical expression involves
DID YOU KNOW more than one operation.

Four less than 3 times a number 3x - 4

ymbols come and symbols go;
S the ones that find the greatest ac-
ceptance are the ones that survive.
Ten more than twice a number 2x + 10
The sum of 5 times a number and 3 5x + 3
The Egyptians used pictorial sym-
Eight times a number decreased by 7 8x - 7
bols: a pair of legs walking forward
for addition or backward for subtrac-
tion. Robert Recorde (1510-1558)
used two parallel lines to represent
"equals" because "no 2 thynges can
be moore equalle." Some symbols
Six more than a number is 10. x + 6 = 10
evolved from abbreviations, such as
the "+" sign, which comes from the Five less than a number is 20. x - 5 = 20
Latin et meaning "and." The evolu- Twice a number decreased by 6 is 12. 2x - 6 = 12
tion of others is less clear. The in- A number decreased by 13 is 6 times the number. x - 13 = 6x
vention of the printing press in the
fifteenth century led to a greater
standardization of symbols already

1. Read the problem carefully at least twice to be sure that you understand it.
2. If possible, draw a sketch to help visualize the problem.
3. Determine which quantity you are being asked to find. Choose a letter to represent
this unknown quantity. Write down exactly what this letter represents.
4. Write the word problem as an equation.
S. Solve the equation for the unknown quantity.
6. Answer the question or questions asked.
7. Check the solution.

This general procedure for solving word problems is illustrated in Examples I

through 4. In these examples, the equations we obtain are mathematical models of the
given situations.

EXAMPLE 1 How Much Can You Purchase?

I Roberto Raynor spent $339.95 on textbooks at the bookstore. In addition to text-

books, he wanted to purchase as many notebooks as possible, but he only has a total
of $350 to spend. If a notebook, including tax, costs $0.99, how many notebooks
can he purchase?

SOLUTION: In this problem, the unknown quantity is the number of notebooks

I Roberto can purchase. Let's select n to represent the number of notebooks he can
purchase. Then we construct an equation using the given information that will
I allow us to solve for n.
DID YOU KNOW

M~O
K tJ III $0.99n = cost for n notebooks at $0.99 per notebook

Cost of textbooks + cost of notebooks = total amount to spend

Heap: t ~~
tJ III
$339.95 + $0.99n $350

339.95 + 0.99n = 350

T he Egyptians as far back as
1650 B.C. had a knowledge of
linear equations. They used the
339.95 - 339.95 + 0.99n = 350 - 339.95
0.99n = 10.05
words aha or heap in place of the
variable. Problems involving linear 0.99n 10.05
-- --
equations can be found in the Rhind 0.99 0.99
Papyrus (See Chapter 4, p. 168).
n ~ 10.15

Therefore, Roberto can purchase 10 notebooks. When we solve the equation, we

obtain n = 10.15 (to the nearest hundredth). Since he cannot purchase a part of a
notebook, only 10 notebooks can be purchased.

I Check: The check is made with the information given in the original problem.

Total amount to spend = cost of textbooks + cost of notebooks

= 339.95 + 0.99( 10)
= 339.95 + 9.90
= 349.85

This result would leave 15 cents change from the $350 he has to spend, which is
Lnot enough to purchase another notebook. Therefore, this answer checks.

Forty hours of overtime must be split among three workers. One worker will be
assigned twice the number of hours as each of the other two. How many hours of
overtime wi]] be assigned to each worker?

SOLUTION: Two workers receive the same amount of overtime, and the third
worker receives twice that amount.

x = number of hours of overtime for the first worker

x = number of hours of overtime for the second worker
2x = number of hours of overtime for the third worker
 x + x + 2x = total amount of overtime
x + x + 2x = 40
4x = 40
x = 10
I Thus, two workers are assigned 10 hours of overtime, and the third worker is as-

I signed 2(10), or 20, hours of overtime. A check in the original problem will verify
that this answer is correct.

Robert Koch wants to fence in a rectangular region in his backyard for his poodle.
He only has 56 ft of fencing to use for the perimeter of the region. What should the
dimensions of the region be if he wants the length to be 4 ft greater than the width?

I SOLUTION: The formula for finding the perimeter of a rectangle is P = 2/ + 2w,

where P is the perimeter, / is the length, and w is the width. A diagram, such as the
lone shown below, is often helpful in solving problems of this type.

Let w equal the width of the region. The length is 4 ft more than the width, so
/= tv+ 4. The total distance around the region P, is 56 ft.
Substitute the known quantities in the formula.

P = 2w + 2/
56 = 2w + 2( w + 4)
56 = 2w + 2w + 8
56 = 4w + 8
48 = 4w
12 = w

In shopping and other daily activities, we are occasionally asked to solve prob-
lems using percents. The word percent means "per hundred." Thus, for example, 7%
means 7 per hundred, or I~O. When I~O is converted to a decimal number, we obtain
0.07. Thus, 7% = 0.07.
Let's look at one example involving percent. (See Section I 1.1 for a more de-
tailed discussion of percent.)
 Peggy McMahon is planning to sell her original paintings at an art show. Determine
the cost of a painting before tax if the total cost of a painting, including an 8% sales
tax, is to be $145.80.

I SOLUTION: We are asked to find the cost of a painting before sales tax.
Let

Cost of a painting before tax + tax on a painting = 145.80

x + 0.08x = 145.80
1.08x = 145.80
1.08x 145.80
-- ---
1.08 ] .08
145.80
x=---
1.08
x = 135

11. 18 decreased by s, divided by 4

12. The sum of 8 and I, divided by 2
1. What is the difference between a mathematical expression
and an equation? 13. 6 less than the product of 5 times y, increased by 3
2. Give an example of a mathematical expression and an ex- 14. The quotient of 8 and y, decreased by 3 times x
ample of a mathematical equation.
In Exercises] 5-26, write an equation and solve.
15. A number decreased by 6 is 5.
16. The sum of a number and 7 is 15.
In Exercises 3-14, write the phrase as a malhematical ex-
17. The difference between a number and 4 is 20.
pression.
18. A number multiplied by 7 is 42.
3. 4 increased by 3 times x
19. Twelve increased by 5 times a number is 47.
4. 6 times x decreased by 2
20. Four times a number decreased by 10 is 42.
5. 5 more than 6 times r
21. Sixteen more than 8 times a number is 88.
6. to times s decreased by 13
22. Six more than five times a number is 7 times the number
7. ]5 decreased by twice r decreased by 18.
8. 6 more than x 23. A number increased by ] 1 is I more than 3 times the num-
9. 2 times m increased by 9 ber.
10. 8 increased by 5 times x 24. A number divided by 3 is 4 less than the number.
25. A number increased by 10 is 2 times the sum of the num- 34. MODELING· Scholarship Donation Each year, Andrea
ber and 3. Choi donates a total of $1000 for scholarships at Mercer
26. The product of 2 and a number decreased by 3 is 4 more County Community College. This year she wants the
than the number. amount she donates for scholarships for liberal arts to be
three times the amount she donates for scholarships for
Problem Solving business. Determine the amount she will donate for each
In Exercises 27-46, set up an equation that can be used to type of scholarship.
solve the problem. Solve the equation and find the desired 35. MODELING - Homeowners Association Cross Creek
valuers}. Townhouses Homeowners Association needs to charge
each homeowner a supplemental assessment to help pay
27. MODELING - Ticket Sales New Hyde Park High School for some unexpected repairs to the townhouses. The asso-
sold 600 tickets to the play The Wiz. The number of tickets ciation has $2000 in its reserve fund that it will use to help
sold to students was three times the number of tickets sold pay for the repairs. How much must the association charge
to nonstudents. How many tickets were sold to students each of the 50 homeowners if the total cost for the repairs
and to nonstudents? is $13,350?
28. MODELING - New Clothing Miguel Garcia purchases two 36. MODELING - Dimensions of a Deck Jim Yuhas is building
new pairs of pants at The Gap for $60. If one pair was $10 a rectangular deck and wants the length to be 3 ft greater
more than the other, how much was the more expensive pair? than the width. What will be the dimensions of the deck if
29. MODELING - Income Tax From 1999 to 2000, there was an the perimeter is to be 54 ft?
11.6% increase in the number of taxpayers filing their 37. MODELING - Floor Area The total floor space in three barns
taxes electronically. If 34.20 million taxpayers filed their is 45,000 ft2. The two smaller ones have the same area, and
taxes electronically in 2000, how many million taxpayers the largest one is three times the area of the smaller ones.
filed their taxes electronically in 1999? a) Determine the floor space for each barn.
30. MODELING - Sales Commission Vinny Raineri receives a b) Can merchandise that takes up 8500 ft2 of floor space
weekly salary of $400 at Abbott's Appliances. He also re- fit into either of the smaller barns?
ceives a 6% commission on the total dollar amount of all 38. MODELING· Average Salary According to the Bureau of
sales he makes. What must his total sales be in a week if Economic Analysis, the average per capita income by state
he is to make a total of $790? in 2000 was highest in Connecticut and lowest in Missis-
31. MODELING· Pet Supplies PetS mart has a sale offering sippi. The average per capita income in Connecticut was
10% off of all pet supplies. If Amanda Miller spent $15.72 $1346 less than two times the per capita income in Missis-
on pet supplies before tax, what was the price of the pet sippi. If the sum of the average per capita income in Con-
supplies he purchased before the discount? necticut and Mississippi is $61,663, what is the average
per capita income in each state?
39. MODELING - Vacation Days According to the World
Tourism Organization, the average number of vacation
days for employees in Italy is 3 more than 3 times the av-
erage number of vacation days for employees in the United
States. The sum of the average number of vacation days in
Italy and in the United States is 55. Determine the average
number of vacation days in each country.
40. MODELING - Car Purchase The Gilberts purchased a car.
If the total cost, including a 5% sales tax, was $14,512,
find the cost of the car before tax.
32. MODELING - Copying Ronnie McNeil pays 8¢ to make a 41. MODELING - Enclosing Two Pens Chuck Salvador has 140
copy of a page at a copy shop. She is considering purchas- ft of fencing in which he wants to build two connecting,
ing a photocopy machine that is on sale for $250, includ- adjacent square pens (see the figure). What will be the di-
ing tax. How many copies would Ronnie have to make in mensions if the length of the entire enclosed region is to be
the copy shop for her cost to equal the purchase price of twice the width?
the photocopy machine she is considering buying?
33. MODELING - Number of CD's Samantha Silverstone and
Josie Appleton receive 12 free compact discs by joining a
compact disc club. How many CD's will each receive if
Josie is to have three times as many as Samantha?
{IJ
\ ,
42. MODELING - Dimensions of a Bookcase A bookcase with 46. MODELING - Truck Rentals The cost of renting a small
three shelves is to be built by a woodworking student. If truck at the U-Haul rental agency is $35 per day plus 20¢
the height of the bookcase is to be 2 ft longer than the a mile. The cost of renting the same truck at the Ryder
length of a shelf and the total amount of wood to be used is rental agency is $25 per day plus 32¢ a mile. How far
32 ft, find the dimensions of the bookcase. would you have to drive in one day for the cost of renting
from U-Haul to equal the cost of renting from Ryder?

Challenge Problems/Group Activities

47. Income Tax Some states allow a husband and wife to file
individual tax returns (on a single form) even though they
have filed a joint federal tax return. It is usually to the tax-
payers' advantage to do so when both husband and wife
work. The smallest amount of tax owed (or the largest re-
fund) will occur when the husband's and wife's taxable in-
comes are the same.
Mr. McAdams's 2003 taxable income was $24,200,
43. MODELING - Laundry Cost The cost of doing the family
and Mrs. McAdams's taxable income for that year was
laundry for a month at a local laundromat is $70. A new $26,400. The McAdams's total tax deduction for the year
washer and dryer cost a total of $760. How many months was $3640. This deduction can be divided between Mr.
would it take for the cost of doing the laundry at the laun- and Mrs. McAdams any way they wish. How should the
dromat to equal the cost of a new washer and dryer? $3640 be divided between them to result in each individ-
ual's having the same taxable income and therefore the
44. MODELING - Health Club Cost A health club is offering
greatest tax refund?
two new membership plans. Plan A costs $56 per month
for unlimited use. Plan B costs $20 per month plus $3 for 48. Write each equation as a sentence. There are many correct
every visit. How many visits to the health club must Doug answers.
Jones make per month for Plan A to result in the same cost a) x + 3 = 13
as Plan B? c) 3x - 8 = 7

45. MODELING - Airfare Rachel James has been told that with
49. Show that the sum of any three consecutive integers is 3
her half-off airfare coupon, her airfare from New York to less than 3 times the largest.
San Diego will be $257.00. The $257.00 includes a 7% tax 50. Auto Insurance A driver education course at the East
on the regular fare. On the way to the airport, Rachel real- Lake School of Driving costs $45 but saves those under 25
izes that she has lost her coupon. What will her regular years of age 10% of their annual insurance premiums until
fare be before tax? they are 25. Dan has just turned 18, and his insurance costs
$600.00 per year.
a) When will the amount saved from insurance equal the
price of the course?
b) Including the cost of the course, when Dan turns 25,
how much will he have saved?

Recreational Mathematics
51. The relationship between Fahrenheit temperature (F) and
Celsius temperature (C) is shown by the formula
9
F = SC + 32. At what temperature will a Fahrenheit
thermometer read the same as a Celsius thermometer?
 In Sections 6.3 and 6.4, we presented many applications of algebra. In this section, we
introduce variation, which is an important tool in solving applied problems.

Many scientific formulas are expressed as variations. A variation is an equation that

relates one variable to one or more other variables through the operations of multipli-
cation or division (or both operations). Essentially there are four types of variation
problems: direct, inverse, joint, and combined variation.
In direct variation, the values of the two related variables increase together or
decrease together; that is, as one increases so does the other, and as one decreases so
does the other.
Consider a car traveling at 40 miles an hour. The car travels 40 miles in 1 hour, 80
miles in 2 hours, and 120 miles in 3 hours. Note that, as the time increases, the dis-
tance traveled increases, and, as the time decreases, the distance traveled decreases.
The formula used to calculate distance traveled is

We say that distance varies directly as time or that distance is directly proportional to
time.
The preceding equation is an example of direct variation.

Direct Variation
If a variable y varies directly with a variable x, then

Circle
Circumference

'" The circumference of a circle, C, is directly proportional to (or varies directly as) its
Radius radius, r; see Fig. 6.S. Write the equation for the circumference of a circle if the

'" constant of proportionality, k, is 271.

 The recommended dosage, d, of the antibiotic drug vancomycin is directly propor-
tional to a person's weight, w.
a) Write this variation as an equation.
b) Find the recommended dosage, in milligrams, for Doug Kulzer, who weighs
192 lb. Assume the constant of proportionality for the dosage is 18.

SOLUTION:
a) d = kw
b) d = 18(192) = 3456
The recommended dosage for Doug Kulzer is 3456 mg.

In certain variation problems, the constant of proportionality, k, may not be

known. In such cases, we can often find it by substituting the given values in the vari-
ation formula and solving for k.

EXAMPLE 3 Finding the Constant of Proportionality

Suppose w varies directly as the square of y. If w is 60 when y is 20, find the con-
stant of proportionality.

SOLUTION: Since w varies directly as the square ofy, we begin with the formula
w = ky2. Since the constant of proportionality is not given, we must find k using
the given information. Substitute 60 for wand 20 for y.

w = ky2

60 = k(20?

60 = 400k

60 400k
- --
400 400
0.15 = k

. EXAMPLE 4 Using the Constant of Proportionality

The length that a spring will stretch, S, varies directly with the force (or weight), F,
attached to the spring. If a spring stretches 4.2 in. when a 60-lb weight is attached,
how far will it stretch when a 30-lb weight is attached?
SOLUTION: We begin with the formula S = kF. Since the constant of proportional-
ity is not given, we must find k using the given information.

S = kF
4.2 = k(60)
4.2
-=k
60
0.07 = k
 S = kF
S = 0.07F
S = 0.07(30)
S = 2.1 in.

A second type of variation is inverse variation. When two quantities vary inversely,
as one quantity increases, the other quantity decreases, and vice versa.
To explain inverse variation, we use the formula, distance = rate· time. If we
solve for time, we get time = distance/rate. Assume the distance is fixed at 100
miles; then

100
Time =--
rate

At 100 miles per hour it would take 1 hour to cover this distance. At 50 miles an hour,
it would take 2 hours. At 25 miles an hour, it would take 4 hours. Note that as the rate
(or speed) decreases, the time increases and vice versa.
The preceding equation can be written

100
t=-
r

This equation is an example of an inverse variation. The time and rate are inversely
proportional. The constant of proportionality in this case is 100.

Inverse Variation
If a variable y varies inversely with a variable x, then

k
y=-
x

Two quantities vary inversely, or are inversely proportional, when as one quantity
increases the other quantity decreases and vice versa. Examples 5 and 6 illustrate in-
verse variation.

'!"" EXAMPLE 5 Inverse Variation in Astronomy

The velocity, v, of a meteorite approaching Earth varies inversely as the square root
of its distance from the center of Earth. Assuming the velocity is 2 miles per second
at a distance of 6400 mi les from the center of Earth, find the equation that expresses
 the relationship between the velocity of a meteorite and its distance from the center

I of Earth.
SOLUTION: Since the velocity of the meteorite varies inversely as the square root of
its distance from the center of Earth, the general form of the equation is

k
y=-
vCt
I To find k, we substitute the given values for Y and d.

k
2 = -\/-64-0-0

k
2 =-
80
(2)(80) =k
160 = k

160
Thus, the formula is Y = ,r.'
L. vd

FEXAMPLE 6 Using the Constant of Proportionality

I Suppose y varies inversely as x. If y = 8 when x = 15, find y when x = 18.

SOLUTION: First write the inverse variation, then solve for k.

k
y=-
x
k
8 =-
15
120 = k

120 120
y = - = - = 6.7 (to the nearest tenth)
x 18

One quantity may vary directly as a product of two or more other quantities. This type
of variation is called joint variation.

Joint Variation
The general form of a joint variation, where y varies directly as x and z, is
 EXAMPLE 7 Joint Variation in Geometry

rThe area, A, of a triangle varies jointly as its base, b, and height, h. If the area of a
triangle is 48 in.2 when its base is 12 in. and its height is 8 in., find the area of a tri-
angle whose base is 15 in. and whose height is 20 in.

SOLUTION: First write the joint variation, then substitute the known values and
solve for k.

A = kbh
48 = k( 12)(8)
48 = k(96)
48
-=k
96

A = kbh
= ~(15)(20)
= 150 in.2

Summary of Variations
Direct
k
y=-
x

Often in real-life situations, one variable varies as a combination of variables. The fol-
lowing examples illustrate the use of combined variations.

EXAMPLE 8 Combined Variation in Engineering

The load, L, that a horizontal beam can safely support varies jointly as the width, w,
and the square of the depth, d, and inversely as the length, l. Express L in terms of
w, d, I, and the constant of proportionality, k.

kwd2
L=--
I

EXAMPLE 9 Pretzel Price, Combined Variation

r The owners of the Colonel Mustard Pretzel Shop find that their weekly sales of
pretzels, S, vary directly with their advertising budget, A, and inversely with their
 pretzel price, P. When their advertising budget is $600 and the price is $1.20, they
sell 6500 pretzels.
a) Write an equation of variation expressing S in terms of A and P. Include the
value of the proportionality constant.

I b) Find the expected sales if the advertising budget is $900 and the pretzel price is
$1.50.

SOLUTION:
a) Since S varies directly as A and inversely as P, we begin with the equation

kA
S=-
P

k(600)
6500 = --
1.20
6500 = 500k
13 = k

13A
Therefore, the equation for the sales of pretzels is S = --.
P
13A
b) S =-
P

l
13(900)
= 7800
1.5 O
They can expect to sell 7800 pretzels.

A varies jointly as Band C and inversely as the square of D. If A = I when

B = 9, C = 4, and D = 6, find A when B = 8, C = 12, and D = 5.

kBC
A=-
D2

We must firSt find the constant of proportionality, k, by substituting the known val-
ues for A, B, C, and D and solving for k.

k(9)(4)
62
36k
36
I = k
 Thus, the constant of proportionality equals I. Now we find A for the corresponding
values of B, C, and D.

kBC
D2
(1)(8)(12) = 96 = 3.84
52 25

Concept/Writing Exercises 8. The time a person spends walking on a treadmill and the
number of calories the person burns
In Exercises 1-4, use complete sentences to answer the
9. The interest earned on an investment and the interest rate
question.
10. The volume of a balloon and its radius
1. Describe inverse variation.
11. A person's speed and the time needed for the person to
2. Describe direct variation. complete the race
3. Describe joint variation. 12. The time required to cool a room and the temperature of
4. Describe combined variation. the room
13. The number of painters hired to paint a house and the time
In Exercises 5-20, use your intuition to determine whether required to paint the house
the variation between the indicated quantities is direct or 14. The number of calories eaten and the amount of exercise
inverse. required to burn off those calories
5. The distance between two cities on a map and the actual 15. The time required to defrost frozen hamburger in a room
distance between the two cities and the temperature of the room
6. The time required to fill a pool with a hose and the volume 16. On Earth, the weight and mass of an object
of water coming from the hose 17. The number of people in the cashier line at the bookstore
and the time required to stand in line
18. The number of books that can be placed upright on a shelf
3 ft long and the width of the books
19. The displacement, in liters, and the horsepower of an
engine
20. The speed of a rider lawn mower and the time it takes to
cut a lawn

21. Name two items that have not been mentioned in this sec-
tion that have a direct variation.
22. Name two items that have not been mentioned in this sec-
tion that have an inverse variation.

Practice the Skills

In Exercises 23-40, (a) write the variation and (b) find the
quantity indicated.

7. The time required to boil water on a burner and the tem- 23. y varies directly as x. Find y when x = 5 and k = 3.
perature of the burner 24. x varies inversely as y. Find x when y = 12 and k = ] 5.
25. m varies inversely as the square of n. Find m when n = 8 square of the distance, d, of the listener from the speaker.
andk=16. If the loudness is 20 dB when the listener is 6 ft from the
26. r varies directly as the square of s. Find r when s = 2 and speaker, find the loudness when the listener is 3 ft from the
k = 13. speaker.

27. R varies inversely as W. Find R when W = 160 and 44. Building a Deck The time, t, it takes to build a deck for a
k = 8. specific house is inversely proportional to the number, n,
of workers building the deck. If it takes two workers 16
28. D varies directly as J and inversely as C. Find D when
hours to build the deck, how many hours will it take for
J = 10, C = 25, and k = 5.
four workers to build the deck?
29. F varies jointly as D and E. Find F when D = 3, E = 10,
and k = 7.
30. A varies jointly as R J and R2 and inversely as the square of
L. Find A when RJ = 120, R2 = 8, L = 5, and k = ~.
31. t varies directly as the square of d and inversely asf If
t = 192 when d = 8 and f = 4, find t when d = 10 and
f = 6.
32. y varies directly as the square root of t and inversely as s.
1fy = 12whent = 36ands = 2, findywhen t = 81
and s = 4.
33. Z varies jointly as Wand Y. If Z = 12 when W = 9 and
Y = 4, find Z when W = 50 and Y = 6. 45. Video Rentals The weekly videotape rentals, R, at
34. y varies directly as the square of R. If y = 4 when R = 4, Busterblock Video vary directly with their advertising
find y when R = 8. budget, A, and inversely with the daily rental price, P.
When the video store's advertising budget is $600 and the
35. H varies directly as L. If H = IS when L = 50, find H
rental price is $3 per day, it rents 4800 tapes per week.
when L = 10.
How many tapes would it rent per week if the store
36. C varies inversely as J. If C = 7 when J = 0.7, find C increased their advertising budget to $700 and raised its
when J = 12. rental price to $3.50?
37. A varies directly as the square of B. If A = 245 when 46. Area and Projection The area, a, of a projected picture on
B = 7, find A when B = 12. a movie screen varies directly as the square of the distance,
38. F varies jointly as Mand M 2 and inversely as the square
J d, from the projector to the screen. If a projector at a dis-
of d. If F = 20 when M J = 5, M 2 = 10, and d = 0.2, tance of 25 feet projects a picture with an area of 100
findFwhenMj = 10,M2 = 20, and d = 0.4. square feet, what is the area of the projected picture when
39. F varies jointly as q land q2 and inversely as the square of the projector is at a distance of 40 feet?
d. If F = 8 when ql = 2, q2 = 8, and d = 4, find F 47. Strength of a Beam The strength, s, of a rectangular beam
when q, = 28, q2 = 12, and d = 2. varies jointly as its width, w, and the square of its depth, d.
40. S varies jointly as I and the square of T. If S = 8 when If the strength of a beam 2 inches wide and 10 inches deep
I = 20 and T = 4, find S when I = 2 and T = 2. is 2250 pounds per square inch, find the strength of a beam
4 inches wide and 12 inches deep.
48. Electric Resistance The electrical resistance of a wire, R,
Problem Solving varies directly as its length, L, and inversely as its cross-
In Exercises 41-49, (a) write the variation and (b) find the sectional area, A. If the resistance of a wire is 0.2 ohm
quantity indicated. when the length is 200 ft and its cross-sectional area is
0.05 in.2, find the resistance of a wire whose length is
41. Resistance The resistance, R, of a wire varies directly as 5000 ft with a cross-sectional area of 0.01 in2.
its length, L. If the resistance of a 30 ft length of wire is
49. Phone Calls The number of phone calls between two
0.24 ohm, find the resistance of a 40 ft length of wire.
cities during a given time period, N, varies directly as the
42. Finding Interest The amount of interest earned on an in- populations p, and P2 of the two cities and inversely to the
vestment, I, varies directly as the interest rate, r. If the in- distance, d, between them. If 100,000 calls are made be-
terest earned is $40 when the interest rate is 4%, find the tween two cities 300 mi apart and the populations of the
amount of interest earned when the interest rate is 6%. cities are 60,000 and 200,000, how many calls are made
43. Speaker Loudness The loudness of a stereo speaker, I, between two cities with populations of 125,000 and
measured in decibels (dB), is inversely proportional to the 175,000 that are 450 mi apart?
SO. a) If y varies directly as x and the constant of proportion- background subject is properly exposed with a flash. Thus
ality is 2, does x vary directly or inversely as y? direct flash will not offer pleasing results if there are any
Explain. intervening objects between the foreground and the
b) Give the new constant of proportionality for x as a vari- subject."
ation of y. If the subject you are photographing is 4 ft from the
51. a) If y varies inversely as x and the constant of proportion- flash and the illumination on this subject is -&. of the light
ality is 0.3, does x vary directly or inversely as y? of the flash, what is the intensity of illumination on an in-
Explain. tervening object that is 3 ft from the flash?
b) Give the new constant of proportionality for x as a vari- 53. Water Cost In a specific region of the country, the amount
ation ofy. of a customer's water bill, W, is directly proportional to the
average daily temperature for the month, T, the lawn area,
A, and the square root of F, where F is the family size, and
Challenge Problems/Group Activities
inversely proportional to the number of inches of rain, R.
52. Photography An article in the magazine Outdoor and In one month, the average daily temperature is 78°F
Travel Photography states, "If a surface is illuminated by and the number of inches of rain is 5.6. If the average fam-
a point-source of light, the intensity of illumination pro- ily of four who has a thousand square feet of lawn pays
duced is inversely proportional to the square of the dis- $72.00 for water for that month, estimate the water bill in
tance separating them. In practical terms, this means that the same month for the average family of six who has
foreground objects will be grossly overexposed if your 1500 ft2 of lawn.

6.6 LINEAR INEQ1TALITIES

The first four sections of this chapter have dealt with equations. However, we often
encounter statements of inequality. The symbols of inequality are as follows.

Symbols of Inequality
a < b means that a is less than b.
a :s b means that a is less than or equal to b.
a > b means that a is greater than b.
a 2: b means that a is greater than or equal to b.

A statement of inequality can be used to indicate a set of real numbers. For exam-
ple, x < 2 represents the set of all real numbers less than 2. Listing all these numbers
is impossible, but some are -2, -1.234, -1, -~, 0, ;:3,5,9.
A method of picturing all real numbers less than 2 is to graph the solution on the
number line. The number line was discussed in Chapter 5.
To indicate the solution set of x < 2 on the number line, we draw an open circle
at 2 and a line to the left of 2 with an arrow at its end. This technique indicates that all
points to the left of 2 are part of the solution set. The open circle indicates that the so-
lution set does not include the number 2.

I ) x<2
5
 To indicate the solution set of x :S 2 on the number line, we draw a closed (or
darkened) circle at 2 and a line to the left of 2 with an arrow at its end. The closed cir-
cle indicates that the 2 is part of the solution.

HI I I I I I I ) x:52
-5 -4 -3 -2 -I 5

Graph the solution set of x :S -2, where x is a real number, on the number line.

SOLUTION: The numbers less than or equal to -2 are all the points on the number
line to the left of -2 and -2 itself. The closed circle at -2 shows that -2 is in-
cluded in the solution set.
•• I I I • I
-5 -4 -3 -2 -I

The inequality statements x < 2 and 2 > x have the same meaning. Note that the
inequality symbol points to the x in both cases. Thus, one inequality may be written in
place of the other. Likewise, x > 2 and 2 < x have the same meaning. Note that the in-
equality symbol points to the 2 in both cases. We make use of this fact in Example 2.

EXAM.PLE 2 Graphing a Less Than Inequality

r Graph the solution set of 3 < x, where x is a real number, on the number line.
SOLUTION: We can restate 3 < x as x > 3. Both statements have identical solu-
tions. Any number that is greater than 3 satisfies the inequality x > 3. The graph

l includes all the points to the right of 3 on the number line. To indicate that 3 is not

partofthe,~;ut;~;,et::wep;"e ono~n CU;le at:3 :. . : ~; A

We can find the solution to an inequality by adding, subtracting, multiplying, or

dividing both sides of the inequality by the same number or expression. We use the
procedure discussed in Section 6.2 to isolate the variable, with one important excep-
tion: When both sides of an inequality are multiplied or divided by a negative number,
the direction of the inequality symbol is reversed.

Solve the inequality - x > 3 and graph the solution set on the number line.
SOLUTION: To solve this inequality, we must eliminate the negative sign in front of
the x. To do so, we multiply both sides of the inequality by -1 and change the di-
rection of the inequality symbol.

Multiply both sides of the inequality by -1 and

change the direction of the inequality symbol.
DID YOU KNOW
( • I I I I I $ I I I ) x<-3
-8 -7 -6 -5 -4 -3 -2 -I 2

EXAMPLE 4 Dividing by a Negative Number

Solve the inequality -4x < 16 and graph the solution set on the number line.
SOLUTION: Solving the inequality requires making the coefficient of the x term 1.
To do so, divide both sides of the inequality by -4 and change the direction of the
inequality symbol.

-4x < 16
-4x 16 Divide both sides of the inequality by -4 and
-->- change the direction of the inequality symbol.
-4 -4

I nthe production of machine parts,

engineers must allow a certain
tolerance in the way parts fit to-
gether. For example, the boring ma-
chines that grind cylindrical open- (I I I Ell I I I
-7 -6 -5 -4 -3 -2 -1
ings in an automobile's engine
block must create a cylinder that al-
lows the piston to move freely up
and down, but still fit tightly enough
to ensure that compression and Solve the inequality 3x - 5 > 13 and graph the solution set on the number line.
combustion are complete. The al-
SOLUTION: To find the solution set, isolate x on one side of the inequality symbol.
lowable tolerance between parts can
be expressed as an inequality. For
example, the diameter of a cylinder
> 13
3x - 5
may need to be no less than 3.383 3x - 5 + 5 > 13 + 5 Add 5 to both sides of the inequality.
in. and no greater than 3.387 in. We
3x> 18
can represent the allowable toler-
ance as 3.383 :=; t :=; 3.387. 3x 18
- >- Divide both sides of the inequality by 3.
3 3
x>6

I • ) x>6
9

Note that in Example 5, the direction of the inequality symbol did not change
when both sides of the inequality were divided by the positive number 3.

Solve the inequality x + 4 < 7, where x is an integer, and graph the solution set on
the number line.

x+4<7
x+4-4<7-4
x < 3
 Since x is an integer and is less than 3, the solution set is the set of integers less
than 3, or {... - 3, -2, -I, 0, I, 2}.To graph the solution set, we make solid dots
at the corresponding points on the number line. The three smaller dots to the left of
-3 indicate that all the integers to the left of -3 are included.

I I·... • • ) x< 3,
-5 -4 -3 -2 -I x an integer

An inequality of the form a < x < his called a compound inequality. Con-
sider the compound inequality -3 < x :s 2, which means that -3 < x and
x :s 2.

Graph the solution set of the inequality - 3 < x :s 2

I a) where x is an integer.
b) where x is a real number.

SOLUTION:
I a) The solution set is all the integers between - 3 and 2, including the 2 but not in-
cluding the -3, or {-2, -1,0,1, 2}.

) -3<x$2,
x an integer

b) The solution set consists of all the real numbers between - 3 and 2, including the
2 but not including the - 3.

l -5
I I
-4
Ell
-3 -2
I I
-I
I
0 •
2
I
4
I
5
) -3 <x $2
•••

x + 3
-4 < --:s 5
2

SOLUTION: To solve a compound inequality, we must isolate the x as the middle

term. To do so, we use the same principles used to solve inequalities.

x + 3
-4 < --:s 5
2

2( -4) < ..2'-(x :- 3) :s 2(5)

-8 < x + 3 :s 10
-8 - 3 < x + 3 - 3 :s 10 - 3
-11<x:s7
"In mathematics the art of posing
problems is easier than that of solving
them." rEXAMPLE 9 Average Grade
A student must have an average (the mean) on five tests that is greater than or equal
to 80% but less than 90% to receive a final grade ofB. Devon's grades on the first
four tests were 98%, 76%, 86%, and 92%. What range of grades on the fifth test
would give him a B in the course?

SOLUTION: The unknown quantity is the range of grades on the fifth test. First con-
struct an inequality that can be used to find the range of grades on the fifth exam.
The average (mean) is found by adding the grades and dividing the sum by the
number of exams.
Let x = the fifth grade. Then

98 + 76 + 86 + 92 + x
5

For Devon to obtain a B, his average must be greater than or equal to 80 but less
than 90.

98 + 76 + 86 + 92 + x
80 :S -------- < 90
5
352 + x
80:s ---< 90
5

5(80) :S .8{ 35~+ x) < 5(90) Multiply the three terms

of the inequality by s.

400 :S 352 + x < 450

400 - 352 :S 352 - 352 + x < 450 - 352
48 :S x < 98

Thus, a grade of 48% up to but not including a grade of 98% on the fifth test will
result in a grade of B.

TIMELY TIP Remember to change the direction of the inequality symbol when
multiplying or dividing both sides of an inequality by a negative number.

Concept/Writing Exercises
1. Give the four inequality symbols we use in this section and
S. Does x > - 3 have the same meaning as - 3 < x?
indicate how each is read.
Explain.
2. a) What is an inequality?
b) Give an example of three inequalities. 6. When graphing the solution set to an inequality on the
3. When solving an inequality, under what conditions do you number line, when should you use an open circle and
need to change the direction of the inequality symbol? when should you use a closed circle?
Practice the Skills p10yee for the years 1997 through 2000 and projected for
2001.
In Exercises 7-24, graph the solution set of the inequality,
where x is a real number, on the number line.
7. x> 6 8. x ~ 9 I $3451
9. x + 4 2:: 7 10. 3x > 9
I $3578
11. -3x ~ 18 12. -4x < 12
x
I $3858
~ < -2
13.6 14. "2 > 4 1$4222

-x
15. - 2:: 3
x
16. - 2:: -4 I $4707
3 2
17. 2x +6 2:: 14 18. 3x + 12 < 5x + 14
19. 4(x - 1) < 6 a) In whjch years was the national average for annual
20. -5(x + 1) + 2x > -3x + 6 health care costs per employee > $4000?
b) In which years was the national average for annual
21. 3(x + 4) - 2 < 3x + 10
health care costs per employee < $3858?
22. -2 ~ x ~ 1 c) In which years was the national average for annual
23. 3 < x - 7 ~ 6 health care costs per employee ~ $4222?
1 x + 4 d) In which years was the national average for annual
24.- < ------
~4 health care costs per employee 2:: $3578?
2 2
46. U.S. Population The following bar graph shows the U.S.
foreign-born population, in mjlhons, for selected years.
In Exercises 25-44, graph the solution set of the inequality,
where x is an integer, on the number line.
u.s. Foreign-born Population in Millions
284
--:-
25. x 2:: 2 26. -3 < x
27. -3x ~ 27 28. 3x 2:: 27
29. x - 2 < 4 30. -5x ~ 15 ~-
x x
31. - ~ -2 32. - 2:: -3 -
3 4 14.2
J12 -
x 2x f-
10.3
33. -- 2:: 3 34. - ~ 4 r-9.2- ~-
6 3 -
35. -11 < -5x +4 f- f-

37. 3(x + 4) 2:: 4x + 13

38. - 2 (x - 1) < 3(x - 4) + 5 1890 1910 1930 1950 1970 1990 2000

39. 5(x + 4) - 6 ~ 2x + 8 Year

40. -3 ~ x < 5
41. 1 > -x > -5
a) In which of the years listed on the bar graph was the
42. - 2 < 2x + 3 < 6 U.S. foreign-born population> 14.2 million?

43. 0.2 ~ ----ro

x-4
~ 0.4
b) In which of the years listed on the bar graph was
U.S. foreign-born population ~ 13.5 million?
the

c) In which of the years listed on the bar graph was the

1 x - 2 1
44. -- < ------ ~ - U.S. foreign-born population 2::28.4 million?
3 12 4 d) In which of the years listed on the bar graph was the
U.S. foreign-born population >19.8 million?
47. Video Rental Movie Mania offers two rental plans. One
Problem Solving has an annual fee and the other has no annual fee. The an-
45. Health Care Costs The following chart shows the national nual membership fee and the daily charge per video for
average for employers for annual health care costs per em- each plan are shown in the table on page 334. Determine
 the maximum number of videos that can be rented for the 54. Speed Limit The minimum speed for vehicles on a high-
no fee plan to cost less than the annual fee plan. way is 40 mph, and the maximum speed is 55 mph. If
Philip Rowe has been driving nonstop along the highway
Rental Plan Yearly Fee Daily Charge per Video
for 4 hr, what range in miles could he have legally traveled?
Annual fee $30 $1.49
55. A Grade of B In Example 9 on page 332, what range of
No fee None $2.99
grades on the fifth test would result in Devon receiving a
48. SaLary PLans Bobby Exler recently accepted a sales posi- grade of B if his grades on the first four tests were 78%,
tion in Portland, Oregon. He can select between the two 64%,88%, and 76%?
salary plans shown in the table. Determine the dollar 56. Tent RentaL The Cuhayoga Community College Planning
amount of weekly sales that would result in Bobby earning Committee wants to rent tents for the spring job fair. Rent-
more by Plan B than by Plan A. a-Tent charges $325 for setup and delivery of its tents.
This fee is charged regardless of the number of tents deliv-
Salary Plan Weekly Salary Commission on Sales
ered and set up. In addition, Rent-a- Tent charges $125 for
Plan A $500 6% each tent rented. If the minimum amount the planning
Plan B $400 8% committee wishes to spend is $950 and the maximum
amount they wish to spend is $1200, determine the mini-
49. Van RentaL The Berrys need to rent a van for their family
mum and maximum number of tents they can rent.
vacation. They can rent a van from Jason's Auto Rentals
for $200 per week with no charge for mileage or from
Challenge Problems/Group Activities
Fred's Fine Autos for $110 per week plus $0.25 per mile.
Determine the distances the Berrys can drive in the van if 57. Painting a House J. B. Davis is painting the exterior of his
the cost of renting from Fred's is to be less than the cost of house. The instructions on the paint can indicate that 1 gal
renting from Jason's. covers from 250 to 400 ft2. The total surface of the house to
be painted is 2750 ft2. Determine the number of gallons of
paint he could use and express the answer as an inequality.

50. Moving Boxes A janitor must move a large shipment of

books from the first floor to the fifth floor. Each box of
books weighs 60 Ib, and the janitor weighs 180 lb. The
sign on the elevator reads, "Maximum weight 1200 lb."
a) Write a statement of inequality to determine the maxi-
mum number of boxes of books the janitor can place on
the elevator at one time. (The janitor must ride in the el-
evator with the books.)
b) Determine the maximum number of boxes that can be 58. Final Exam Teresa's five test grades for the semester are
moved in one trip. 86%, 74%, 68%, 96%, and 72%. Her final exam counts
51. Price of a MeaL After Mrs. Franklin is seated in a restaurant, one-third of her final grade. What range of grades on her
she realizes that she has only $19.00. If she must pay 7% tax final exam would result in Teresa receiving a final grade of
and wants to leave a 15% tip on the price of the meal before B in the course? (See Example 9.)
tax, what is the price range of meals that she can order? 59. A student multiplied both sides of the inequality
52. Making a Profit For a business to realize a profit, its rev- . -* x s; 4 by - 3 and forgot to reverse the direction of the
enue, R, must be greater than its costs, C; that is, a profit inequality symbol. What is the relation between the stu-
will only result if R > C (the company breaks even when dent's incorrect solution set and the correct solution set? Is
R = C). A book publishing company has a weekly cost any number in both the correct solution set and the stu-
equation of C = 2x + 2000 and a weekly revenue equa- dent's incorrect solution set? If so, what is it?
tion of R = 12x, where x is the number of books pro-
duced and sold in a week. How many books must be sold Internet/Research Activity
weekly for the company to make a profit? 60. Find a newspaper or a magazine article that contains the
53. Finding Velocity The velocity, v, in feet per second, t sec mathematical concept of inequality.
after a tennis ball is projected directly upward is given by a) From the information in the article write a statement of
the formula v = 84 - 32t. How many seconds after be- inequality.
ing projected upward will the velocity be between 36 ft/sec b) Summarize the article and explain how you arrived at
and 68 ft/sec? the inequality statement in part (a).
6.7 GRAPHING LINEAR EQl1ATIONS
In Section 6.2, we solved equations with a single variable. Real-world problems, how-
ever, often involve two or more unknowns. For example, the profit, p, of a company
may depend on the amount of sales, s; or the cost, c, of mailing a package may depend
on the weight, w, of the package. Thus, it is helpful to be able to work with equations
with two variables (for example, x + 2y = 6). Doing so requires understanding the
Cartesian (or rectangular) coordinate system, named after the French mathemati-
cian Rene Descartes (1596-1650).
The rectangular coordinate system consists of two perpendicular number lines
(Fig. 6.6). The horizontal line is the x-axis, and the vertical line is the y-axis. The
point of intersection of the x-axis and y-axis is called the origin. The numbers on the
axes to the right and above the origin are positive. The numbers on the axes on the left
and below the origin are negative. The axes divide the plane into four parts: the first,
second, third, and fourth quadrants.

5
4
2nd quadrant 3
2
I /Origin (0,0)

-5-4-3-2-1 -)
1 2 3 4 5
ollywood studio executives use
H four "quadrants" to divide up
the movie-going audience. Men 25
3rd quadrant
-2
-3
-4
years and older, men younger than -5
25, women 25 years and older, and
women younger than 25 are the age
groups represented by Hollywood's
four quadrants. If a studio produces a We indicate the location of a point in the rectangular coordinate system by means
movie that appeals to all four quad- of an ordered pair of the form (x, y). The x-coordinate is always placed first and the y-
rants, it is sure to have a hit movie. If coordinate is always placed second in the ordered pair. Consider the point illustrated
the movie appeals to none of the four
in Fig. 6.7. Since the x-coordinate of the point is 5 and the y-coordinate is 3, the or-
quadrants, the movie is sure to fail. A
dered pair that represents this point is (5, 3).
challenge for studio executives is to
determine the core quadrant and then
try to make sure no other movie
geared toward the same quadrant de-
buts at the same time. One of the
biggest hit movies of the summer of
2001 was Shrek, a movie that ap-
-----(5,3) ..
I
I
pealed to all four quadrants. On the I
other hand, the target or core audience -5-4-3-2-t1 1 234 5
for the movie A./. Artificial Intelli-
-2
gence, also released in the summer of
-3
2000, was unclear.The logo and pre- -4
views for A.I., featured the child star, -5
whereas everything else suggested a
moviemore appropJiatefor adults. As
a result, the marketing campaign con-
fused both audiences, and A.l. was The origin is represented by the ordered pair (0, 0). Every point on the plane can
much more a critical success than a be represented by one and only one ordered pair (x, y), and every ordered pair (x, y)
box office success.
represents one and only one point on the plane.
 EXAMPLE 1 Plotting Points
PROFILE IN Plot the points A( -2,4), B(3, -4), C(6,0), D( 4, 1), and £(0, 3).
MATHEMATICS SOLUTION: Point A has an x-coordinate of -2 and a y-coordinate of 4. Project a
, vertical line up from -2 on the x-axis and a horizontal line to the left from 4 on the
RENE DESCARTES y-axis. The two lines intersect at the point denoted A (Fig. 6.8). The other points are
plotted in a similar manner.

A (-2, 4) 5
"-4
I 3 £(0,3)
I
I 2
D(4, I)
I 1
e C(6,0)
-3 -2 - '-I I 2 3 4 5 6 7
The Mathematician and the Fly
-2
ccording to legend, the French
A mathematician and philoso-
pher Rene Descartes (1596-1650)
-3
-4 e8(3, -4)
-5
did some of his best thinking in
bed. He was a sickly chiid, and so
the Jesuits who undertook his edu-
cation allowed him to stay in bed
each morning as long as he liked.
This practice he carried into adult-
The points, A, B, and C are three vertices of a parallelogram with two sides parallel
hood, seldom getting up before
noon. One morning as he watched a
to the x-axis. Plot the three points below and determine the coordinates of the
fly crawl about the ceiling, near the fourth vertex, D.
corner of his room, he was struck
with the idea that the fly's position
could best be described by the con-
necting distances from it to the two
SOLUTION: A parallelogram is a figure that has opposite sides that are of equal
adjacent walls. These became the
coordinates of his rectangular coor- length and are parallel. (Parallel lines are two lines in the same plane that do not in-
dinate system and were appropri- tersect.) The horizontal distance between points Band Cis 5 units (see Fig. 6.9).
ately named after him (Cartesian Therefore, the horizontal distance between points A and D must also be 5 units.
coordinates) and not the fly. This problem has two possible solutions, as illustrated in Fig. 6.9. In each figure,
we have indicated the given points in red.

6 6
5 8(2,4) 5 units 8(2,4) 5 units
'" ,.--------A---- C (7, 4) D(-4,2) '" ,.--------A---- C (7 , 4)

A(l,2)~ /
'---------y----
5 units
7 D (6, 2)
I
Graphing Linear Equations by Plotting Points
Consider the following equation in two variables: y = x + 1. Every ordered pair that
makes the equation a true statement is a solution to, or satisfies, the equation. We can
mentally find some ordered pairs that satisfy the equation y = x + 1 by picking
some values of x and solving the equation for y. For example, suppose we let x = I;
then y = I + 1 = 2. The ordered pair (I, 2) is a solution to the equation y = x + I.
We can make a chart of other ordered pairs that are solutions to the equation.

x y Ordered Pair

I 2 (1,2)
2 3 (2,3)
3 4 (3,4)
4.5 5.5 (4.5,5.5)
-3 -2 (-3, -2)

How many other ordered pairs satisfy the equation? Infinitely many ordered pairs sat-
isfy the equation. Since we cannot list all the solutions, we show them by means of a
graph. A graph is an illustration of all the points whose coordinates satisfy an equa-
tion.
The points (1, 2), (2, 3), (3, 4), (4.5, 5.5), and (- 3, -2) are plotted in Fig. 6.10.
With a straightedge we can draw one line that contains all these points. This line,
when extended indefinitely in either direction, passes through all the points in the
plane that satisfy the equation y = x + 1. The arrows on the ends of the line indicate
that the line extends indefinitely.

All equations of the form ax + by = c, a =f:. 0, b =f:. 0, will be straight lines

when graphed. Thus, such equations are called linear equations in two variables. The
exponents on the variables x and y must be 1 for the equation to be linear. Since only
two points are needed to draw a line, only two points are needed to graph a linear
equation. It is always a good idea to plot a third point as a checkpoint. If no error has
been made, all three points will be in a line, or collinear. One method that can be used
to obtain points is to solve the equation for y, substitute values for x, and find the cor-
responding values of y.
 EXAMPLE 3 Graphing an Equation by Plotting Points
DID YOU KNOW
Graph y = 2x + 4.
qr~ eaicu.ldorr SOLUTION: Since the equation is already solved for y, select values for x and find
the corresponding values for y. The table indicates values arbitrarily selected for x
(
•.1"
,/
and the corresponding values for y. The ordered pairs are (0, 4), (1, 6), and ( -2,0) .
The graph is shown in Fig. 6.11.
,.' J

/
(/ ..
y
..•.. x y
/ 8
7
0 4
6
I 6
5 ~ Y intercept
-2 0 4 (0.4)
n page 308, we illustrated the
O graph of an exponential equa-
tion we obtained from the screen (or
y=2x+4

window) of a TI-83 Plus graphing

1 2 3 4 5 x
calculator. Figure 6.12 shows the
graph of the equation y = 2x + 4
as illustrated on the screen of that
calculator.
Graphing calculators are being used
more and more in many mathemat-
ics courses. Other makers of graph-
ing calculators include Casio, Sharp,
and Hewlett-Packard. Graphing cal-
culators can do a great many more
things than a scientific calculator
can. If you plan on taking additional I. Solve the equation for y.
mathematics courses, you may find 2. Select at least three values for x and find their corresponding values of y.
that a graphing calculator is required
3. Plot the points.
for those courses.
4. The points should be in a straight line. Draw a line through the set of points and
place arrow tips at both ends of the line.

In step 4 of the procedure, if the points are not in a straight line, recheck your cal-
culations and find your error.

Graphing by Using Intercepts

Example 3 contained two special points on the graph, (-2,0) and (0, 4). At these
points, the line crosses the x-axis and the y-axis, respectively. The ordered pairs
( - 2, 0) and (0, 4) represent the x-intercept and the y-intercept, respectively. Another
method that can be used to graph linear equations is to find the x- and y-intercepts of
the graph.

Finding the x- and y- Intercepts

To find the x-intercept, set y = 0 and solve the equation for x.
To find the y-intercept, set x = 0 and solve the equation for y.
 An equation may be graphed by finding the x- and y-intercepts, plotting the inter-
DID YOU KNOW cepts, and drawing a straight line through the intercepts. When graphing by this
method, you should always plot a checkpoint before drawing your graph. To obtain a
checkpoint, select a nonzero value for x and find the corresponding value of y. The
checkpoint should be collinear with the x- and y-intercepts.

,...EXAMPLE 4 Graphing Using Intercepts

I Graph 2x - 4y = 8 by using the x- and y-intercepts.

SOLUTION: To find the x-intercept, set y = 0 and solve for x.

2x - 4y = 8
2x - 4(0) = 8
2x = 8
x = 4

2x - 4y = 8
2(0) - 4y = 8
-4y = 8
rids have long been used in
G mapping. In archaeological
digs, a rectangular coordinate sys-
y = -2

The y-intercept is (0, -2). As a checkpoint, try x = 2 and find the corresponding
tem may be used to chart the loca-
value for y.
tion of each find.
2x - 4y = 8
2(2) - 4y = 8
4 - 4y = 8
-4y = 4
y = -1

Figure 6.13

I Since all three points in Fig. 6.13 are collinear, draw a line through the three points
to obtain the graph. .•.
L.
 Y Horizontal change Slope
(x2-x,)
~ Another useful concept when you are working with straight lines is slope, which is a
Vel1ical
change
Y2 {f - - - - B (x2' Y2) measure of the "steepness" of a line. The sLope of a Line is a ratio of the vertical
change to the horizontal change for any two points on the line. Consider Fig. 6.14.
(Y2 - Y,) A(xl,yl)
Point A has coordinates (x" y,), and point B has coordinates (X2, Y2)' The vertical
change between points A and B is Y2 - Y" and the horizontal change between points
A and B is X2 - XI. Thus, the slope, which is often symbolized with the letter m, can
be found as follows.

vertical change
Slope = ------
horizontal change
Y2 - YI

The Greek capital letter delta, ~, is often used to represent the words "the change
in." Therefore, slope may be defined as

~Y
m=-
~x

A line may have a positive slope, a negative slope, zero slope, or the slope may be
undefined, as indicated in Fig. 6.15. A line with a positive slope rises from left to
right, as shown in Fig. 6.15(a). A line with a negative slope falls from left to right, as
shown in Fig. 6.15(b). A horizontal line, which neither rises nor falls, has a slope of
zero, as shown in Fig. 6.15(c). Since a vertical line does not have any horizontal
change (the x value remains constant) and since we cannot divide by 0, the slope of a
vetticalline is undefined, as shown in Fig. 6.15(d).

Positive slope (117 > 0) Negative slope (117 < 0) Zero slope (117 = 0) Slope is undefined
W ~ ~ ~

EXAMPLE 5 Finding the SLope of a Line

I Determine the slope of the line that passes through the points ( -1, - 3) and (1, 5).

I SOLUTION: Let's begin by drawing a sketch, illustrating the points and the line. See
Fig. 6.16(a) on page 341.
We will let (Xl, YI) be (-1, -3) and (X2, Y2) be (1,5). Then

Y2 - YI (-3) = 5+ 3 = ~ = ~ = 4
Sl ope =
X2 - XI (-1) 1+12 I
 The slope of 4 means that there is a vertical change of 4 units for each horizon-
DID YOU KNOW tal change of I unit; see Fig. 6.16(b). The slope is positive, and the line rises from
left to right. Note that we would have obtained the same results if we let (xj, YI) be
(I, 5) and (Xl> Y2) be ( -1, - 3). Try this now and see.

5 -
Horizontal 4
chancre'
I uni~ . '\~

-5 -4 -3 -2 -~
I
Vertical /: 2
lthough we may not think
A much about it, the slope of a
line is something we are altogether
change:
4 units
-4
3

-5
familiar with. You confront it every
time you run up the stairs, late for
class, moving 8 inches horizontally
for every 6 inches up. The 2002
Olympic gold medalist Simon Am-
mann of Switzerland is familiar with
the concept of slope. He speeds
down a steep 120 meters of a ski
ramp at speeds of over 60 mph be- Graphing Equations by Using the Slope
fore he takes flight.
and y- Intercept

In the equation Y = mx + b, b represents the value of Y where the graph of the

equation Y = mx + b crosses the y-axis.
Consider the graph of the equation y = 3x + 4, which appears in Fig. 6.17. By
examining the graph, we can see that the y-intercept is (0, 4). We can also see that the
graph has a positive slope, since it rises from left to right. Since the vertical change is
3 units for every 1 unit of horizontal change, the slope must be t or 3.
We could graph this equation by marking the y-intercept at (0, 4) and then mov-
ing up 3 units and to the right I unit to get another point. If the slope were - 3,which
means ~3, we could start at the y-intercept and move down 3 units and to the right I
unit. Thus, if we know the slope and y-intercept of a line, we can graph the line.
 To Graph Equations by Using the Slope and y-Intercept
1. Solve the equation for y to place the equation in slope-intercept form.
2. Determine the slope and y-intercept from the equation.
3. Plot the y-intercept
4. Obtain a second point using the slope.
5. Draw a straight line through the points.

-5 -4 -3 -2 -I
-I
-2
-3
-4
-5
-6
r EXAMPLE 6
Graph y = -3x +
Graphing an Equation Using the Slope and y-Intercept
1 using the slope and y-intercept
-3
SOLUTION: The slope is - 3 or - and the y-intercept is (0, 1). Plot (0, 1) on the

I
-7

y-axis. Then plot the next point b; moving down 3 units and to the right 1 unit
(see Fig. 6.18). A third point has been plotted in the same way. The graph of

L y = -3x + I is the line drawn through these three points.

EXAMPLE 7 Write an Equation in Slope-Intercept Form

r a) Write 3x - 5y
b) Graph the equation.
= 10 in slope-intercept form.

SOLUTION:
a) To write 3x - 5y = 10 in slope-intercept form, we solve the given equation for y.

-5 -4 -3 -2 -I
'" -I
3x - 5y = 10
1'= 2.1'-2 -
. 5) IV-
-3
3x - 3x - 5y = - 3x + 10
-4 -5y = -3x + 10
-5 -5y -3x + 10
-5 -5
-3x 10 3
y=-+- y = -x - 2
-5 -5 5
Thus, in slope-intercept form, the equation is y = ~x - 2.
b) The y-intercept is (0, -2) and the slope is~. Plot a point at (0, -2) on the y-
axis, then move up 3 units and to the right 5 units to obtain the second point (see
Fig. 6.19). Draw a line through the two points. A

-5 -4 -3 -2 -I
-[ EXAMPLE 8 Determine the Equation of a Line from Its Graph

r
-2
-3 Determine the equation of the line in Fig. 6.20.
-4
-5 SOLUTION: If we determine the slope and the y-intercept of the line, then we can
write the equation using slope-intercept form, y = mx + b. We see from the graph
that the y-intercept is (0, 2); thus, b = 2. The slope of the line is negative because
 I the graph falls from left to right. The change in y is 2 units for every 3-unit change
in x. Thus, 111, the slope of the line, is -~.

y = I11X + b
2
y = --x +2
3

EXAMPLE 9 Horizontal and Vertical Lines

I In the Cartesian coordinate system, graph (a) y

SOLUTION:
= 2 and (b) x = -3.

a) For any value of x, the value of y is 2. Therefore, the graph will be a horizontal
line through y = 2 (Fig. 6.21).
b) For any value of y, the value of x is -3. Therefore, the graph will be a vertical
I line through x = - 3 (Fig. 6.22).
Note that the graph of y = 2 has a slope of O. The slope of the graph of x = - 3 is
Lundefined.
In graphing the equations in this section, we labeled the horizontal axis the x-axis
-6-5-4- -2-'-1 and the vertical axis the y-axis. For each equation, we can determine values for y by
-2
substituting values for x. Since the value of y depends on the value of x, we refer to y
as the dependent variable and x as the independent variable. We label the vertical
axis with the dependent variable and the horizontal axis with the independent vari-
able. For the equation C = 3n + 5, the C is the dependent variable and n is the inde-
pendent variable. Thus, to graph this equation, we label the vertical axis C and the
horizontal axis n.
In many graphs, the values to be plotted on one axis are much greater than the
values to be plotted on the other axis. When that occurs, we can use different scales on
the horizontal and the vertical axes, as illustrated in Examples 10 and 11. The next
two examples illustrate applications of graphing.

rEXAMPLE 10

S 20
I The Professional Patio Company installs brick patios. The area of brick, a, in square
feet, the company can install in t hours can be approximated by the formula a = St.

"
~
« 10 I a) Graph a = 5t, for t :::;6.
b) Use the graph to estimate the area of brick the company can install in 4 hours.

o 2 4 6t SOLUTION:
Time (hr)
I a) Since a = 5t is a linear equation, its graph will be a straight line. Select three val-
ues for t, find the corresponding values for a, and then draw the graph (Fig. 6.23).
a = 5t
Let t = 0, a = 5(0) = 0 o 0
Let t = 2, a = 5(2) = 10 2 10
Let t = 6, a = 5(6) = 30 6 30
 b) By drawing a vertical line from t = 4 on the time axis up to the graph and then
drawing a horizontal line across to the area axis, we can determine that the area
L installed in 4 hours is 20 ft2.

Jonathan Cwirko owns a small business that manufactures compact discs. He be-
lieves that the profit (or loss) from the compact discs produced can be estimated by
the formula P = 3.55 - 200,000, where 5 is the number of compact discs sold.
a) Graph P = 3.55 - 200,000, for 5 :::; 500,000 compact discs.
b) From the graph, estimate the number of compact discs that must be sold for the
company to break even.
c) If the profit from selling compact discs is $1 million, estimate the number of
compact discs sold.

SOLUTION:
a) Select values for 5 and find the corresponding values of P.

s
816
o -200,000 g 14
100,000 150,000 g 12
;;;;; 10
500,000 1,550,000 5 8
<E 6
~ 4
2

b) On the graph (Fig. 6.24), note that the break-even point is about 0.6, or 60,000
compact discs.
c) We can obtain the answer by drawing a horizontal line from 10 on the profit
axis. Since the horizontal line cuts the graph at about 3.4 on the 5 axis, approxi-
l mately 340,000 compact discs were sold.

Concept/Writing Exercises 7. a) In which quadrant is the point (1,4) located?

1. What is a graph? b) In which quadrant is the point (-2,5) located?
2. Explain how to find the x-intercept of a linear equation. 8. What is the minimum number of points needed to graph a
linear equation?
3. Explain how to find the y-intercept of a linear equation.
4. What is the slope of a line?
S. a) Explain in your own words how to find the slope of a
line between two points.
b) Based on your explanation in part (a), find the slope of
the line through the points (6, 2) and (-3,5). 9.(-3,2) 10. (2, 3) 11.(-5,-1)
6. Describe the three methods used to graph a linear equation 12. (4,0) 13. (0,2) 14. (0,0)
in this section. 15. (0, -5) 16. (3t 4~)
 59. 3x + y = 6 60. 4x - 2y = 12
17. (5, 1) 18. (0, -3) 19. (-6, -1) 61. 2x = -4y - 8 62. y = 4x +4
20. (1,0) 21. (-3,0) 22. (-3, I) 63.y = 3x +5 64. 3x + 6y = 9

23. (4, -1) 24. (4.5,3.5) 65. 3x - y = 5 66. 4y = 2x + 12

In Exercises 25-34 (indicated on Fig. 6.25), write the coor- In Exercises 67-76, find the slope of the line through the
dinates of the corresponding point. given points. If the slope is undefined, so state.
y 67. (3,7) and (10, 21) 68. (4, I) and (1, 4)
5 69. (2,6) and (-5, -9) 70. (-5,6) and (7, -9)
26
• 4
34 71. (5,2) and (-3,2) 72. (-3, -5) and (-1, -2)

is
>5
1
•33
• 73. (8, -3) and (8, 3)
75. (-2,3) and (1, -I)
74. (2,6) and (2, -3)
76. (-7, -5) and (5, -6)
27 32
-5 -4 -3 -2 -I I 2 3 4 5 x
-1 In Exercises 77-86, graph the equation using the slope and
-2 y-intercept, as in Examples 6 and 7.
•29 -3
-4
30
•31 77. y = x +3 78. y = 3x +2
-5 79. y = -x - 4 80. y = -2x + 1
81. y = -ix +2 82. y = -x - 2
Figure 6.25
83.7y =4x - 7 84. 3x + 2y = 6
In Exercises 35-42, determine which ordered pairs satisfy 85. 3x - 2y + 6 = 0 86. 3x + 4y - 8 = 0
the given equation.
35. 3x +y =7 ( I, 3) (1,4) (-1,10)
36. 4x - y = 4 (0, -4) (1,0) (2,-3)
37. 2x - 3y = 10 (5,0) (0,3) (0, -~)
38. 3y = 4x +2 (2, l) ( 1, 2) (0,5)
39. 7y = 3x - 5 (1, -1), (-3, -2), (2,5)
x
40'"2 + 3y = 4 (0, ~), (8,0), (10,-2)
-4 -3 -2 -)
-)
x 3y
41. - +- = 2 (0, ~), (1,7), (4,0) -2
2 4
42. 2x - 5y = -7 (2, 1), (-I, 1), (4,3)

In Exercises 43-46, graph the equation and state the slope

of the line if the slope exists (see Example 9).

In Exercises 47-56, graph the equation by plotting points,

as in Example 3.
47.y = x +3 48. y = x - 2
49. y = 2x - 1 50. y = -x +4
51. y + 3x = 6 52. y - 4x = 8
53. y = ~x +4 54. 3y = 2x - 3
55.2y = -x +6 56. y = -~x

In Exercises 57-66, graph the equation, using the x- and

y-intercepts, as in Example 4.
 101. Photo Processing The charge, C, for processing a roll of
35-millimeter (mm) film onto a picture compact disc at
Costco's I Hour Photo is $8.95 plus $0.33 per picture, or
C = 8.95 + 0.33n, where n is the number of pictures
printed.

Problem Solving
In Exercises 91 and 92, points A, 8, and C are three ver-
tices of a rectangle (the points where two sides meet). Plot
the three points. (a) Find the coordinates of the fourth a) Draw a graph of the cost of processing film for up to
point, D, to complete the rectangle. (b) Find the area of the and including 36 pictures.
rectangle; use A = lw. b) From the graph, estimate the cost of processing a roll
of 35 mm film containing 20 pictures.
91. A ( -1, 4), 8(4,4), C(4,2)
c) If the total cost of processing a roll of 35 mm film is
92. A( -4,2), 8(7,2), C(7,8) $20.83, estimate the number of pictures.
In Exercises 93 and 94, points A, 8, and C are three ver- 102. Earning Simple Interest When $1000 is invested in a
tices of a parallelogram with sides parallel to the x-axis. savings account paying simple interest for a year, the in-
Plot the three points. Find the coordinates of the fourth terest, i in do\lars, earned can be found by the formula
i = 1000r, where r is the rate in decimal form.
point, D, to complete the parallelogram. Note: There are
a) Graph i = 1000r, for r up to and including a rate of
two possible answers for point D.
15%.
93. A(3, 2), 8(5, 5), C(9,5) b) If the rate is 4%, what is the simple interest?
94. A( -2,2), 8(3,2), C(6,-1) c) If the rate is 6%, what is the simple interest?

In Exercises 95-98, for what value of b will the line joining In Exercises 103 and 104, a set of points is plotted. Also
the points P and Q be parallel to the indicated axis? shown is a straight line through the set of points that is
called the line of best fit (or a regression line, as will be
95.P(-1,3), Q(4,b); x-axis
discussed in Chapter 13, Statistics.)
96. P(5, 6), Q(b, -2); y-axis
103. Determining a Test Grade The graph shows the hours
97. P(3b - 1,5), Q(8,4); y-axis
studied and the test grades on a biology test for six stu-
98. P(-6, 2b + 3), Q(7, -I); x-axis dents. (The two points indicated on the line do not repre-
99. Selling Chocolates Ryan Stewart sells chocolate on the In- sent any of the six students.) The line of best fit, the red
ternet. His monthly profit, p, in dolJars, can be estimated line on the graph, can be used to approximate the test
by p = l5n - 300, where n is the number of dozens of grade the average student receives for the number of
chocolates he sells in a month. hours he or she studies.
a) Graph p = l5n - 300, for n S; 60.
b) From the graph, estimate his profit if he sells 40 dozen
chocolates in a month.
c) How many dozens of chocolates must he sell in a
month to break even?
100. Hanging Wallpaper Tanisha Vizquez owns a wallpaper
hanging business. Her charge, C, for hanging wallpaper is
$40 plus $0.30 per square foot of wallpaper she hangs, or
C = 40 + 0.30s, where s is the number of square feet of
wallpaper she hangs.
a) Graph C = 40 + 0.30s, for s s; 500.
b) From the graph, estimate her charge if she hangs 300
square feet of wallpaper.
c) If her charge is $70, use the equation for C to deter- 234
mine how many square feet of wallpaper she hung. Hours studied
 a) Determine the slope of the line of best fit using the 2000 would be represented by 30. Using the ordered pairs
two points indicated. (0,40) and (30, 24),
b) Using the slope determined in part (a) and the y-inter- a) determine the slope of the dashed line.
cept, (0, 53), determine the equation of the line of best b) determine the equation of the dashed line using
fit. (0,40) as the y-intercept.
c) Using the equation you determined in part (b), deter- c) Using the equation you determined in part (b), deter-
mine the approximate test grade for a student who mine the percent of married householders with chil-
studied for 3 hours. dren in 1985, which would be represented by 15.
d) Using the equation you determined in part (b), deter- d) Using the equation you determined in part (b), deter-
mine the amount of time a student would need to mine the year in which the percent of married house-
study to receive a grade of 80 on the biology test. holders with children was 30.
104. Determining the Number of Defects The graph shows
the daily number of workers absent from the assembly
line at J. B. Davis Corporation and the number of defects 40%
coming off the assembly line for 8 days. (The two points
'" '-...:
~~~
indicated on the line do not represent any of the 8 days.)
The line of best fit, the blue line on the graph, can be
used to approximate the number of defects coming off
the assembly line per day for a given number of workers
-
~~~
--...: ~~~
~-
24%
absent.
a) Determine the slope of the line of best fit using the
two points indicated.
b) Using the slope determined in part (a) and the
y-intercept, (0, 9), determine the equation of the line
of best fit.
106. Book SaLes The green graph shows book publishers' net
c) Using the equation you determined in part (b), deter-
sales, in millions of dollars, for trade, mass market, pro-
mine the approximate number of defects for a day if 3
fessional, educational, and university press publishers.
workers are absent.
The red dashed straight line can be used to approximate
d) Using the equation you determjned in part (b), ap-
the book publishers' net dollar sales. If we let 0 represent
proximate the number of workers absent for a day if
1994, I represent 1995,2 represent 1996, and so on, then
there are 17 defects that day.
2003 would be represented by 9. Using the ordered pairs
(0, 17,000) and (9, 25,000),
Daily Number of Defects at a) determine the slope of the dashed line.
J.B. Davis Corporation b) determine the equation of the dashed line using
(0, 17,000) as the y-intercept of the graph.
20 c) Using the equation you determined in part (b), deter-
19 mine the net dollar sales in 1998, which would be
18 represented with year 4.
'"t> 17 d) Using the equation you determined in part (b), deter-
<E 16 mine the year that net sales were $20,000 (in millions).
G)
0 15
"-0 14
'""'
G)
.n 13
E $30,000
i- 12
11 25,000
~
10
9
:§= 20,000 ---
·s 15,000
0 2 3 4 5 5
Workers Absent •..'" 10,000
'i
ff1

5,000
105. Married HousehoLders with ChiLdren The blue graph o
shows the percent of married householders with children. 1994 '95 '96 '97 '98 '99 '00 '01 '02 '03
The red dashed straight line can be used to approximate Year Projections
the percent of married householders with children. If we
let 0 represent 1970, 10 represent 1980, and so on, then
Challenge Problems/Group Activities 108. In which quadrants will the set of points that satisfy the
equation x + y = 1 lie? Explain.
107. a) Two lines are parallel when they do not intersect no
matter how far they are extended. Explain how you
can determine, without graphing the equations,
Internet/Research Activity
whether two equations will be parallel lines when
graphed. 109. Rene Descartes is known for his contributions to
b) Determine whether the graphs of the equations algebra. Write a paper on his life and his contributions
2x - 3y = 6 and 4x = 6y + 6 are parallel lines. to algebra.

6.8 LINEAR INEQlJALITIES IN TWO VARIABLES

In Section 6.6, we introduced linear inequalities in one variable. Now we will intro-
duce linear inequalities in two variables. Some examples of linear inequalities in two
variables are 2x + 3y ~ 7, x + 7y 2: 5, and x - 3y < 6.
The solution set of a linear inequality in one variable may be indicated on a num-
ber line. The solution set of a linear inequality in two variables is indicated on a coor-
dinate plane.
An inequality that is strictly less than «) or greater than (» will have as its
solution set a half-plane. A half-plane is the set of all the points on one side of a line.
An inequality that is less than or equal to (~) or greater than or equal to (2:) will
have as its solution set the set of points that consists of a half-plane and a line. To in-
dicate that the line is part of the solution set, we draw a solid line. To indicate that the
line is not part of the solution set, we draw a dashed line.

To Graph Inequalities in Two Variables

1. Mentally substitute the equal sign for the inequality sign and plot points as if you
were graphing the equation.
2. If the inequality is < or >, draw a dashed line through the points. If the inequal-
ity is ~ or 2:, draw a solid line through the points.
3. Select a test point not on the line and substitute the x- and y-coordinates into the
inequality. If the substitution results in a true statement, shade in the area on the
same side of the line as the test point. If the test point results in a false statement,
shade in the area on the opposite side of the line as the test point.

~... 5
•..•.. 4
•.. x + 2)' =4
•..•..•..•..3 (0,2) EXAMPLE 1 Graphing an Inequality
"2...... (2, I)
1 ••••••••••• (4,0)
Draw the graph of x + 2y < 4.
-5 -4 -3 -2 -J 1 2 3 4 -s........ x SOLUTION: To obtain the solution set, start by graphing x + 2y = 4. Since the
-I
-2
original inequality is strictly "less than," draw a dashed line (Fig 6.26). The dashed
-3 line indicates that the points on the line are not part of the solution set.
-4 I The line x + 2y = 4 divides the plane into three parts, the line itself and two
-5 half-planes. The line is the boundary between the two half-planes. The points in
one half-plane will satisfy the inequality x + 2y < 4. The points in the other half-
I plane will satisfy the inequality x + 2y > 4.
 To determine the solution set to the inequality x + 2y < 4, pick any point on
I the plane that is not on the line. The simplest point to work with is the origin, (0, 0).
~... 5
Substitute x = 0 and y = 0 into x + 2y < 4.
............4
•.. x + 2y =4
•..•..•..; (0,2) + 2y < 4
x
2 •..•.. (2, 1)
I •..••..•.. (4,0)
Is 0 + 2 (0) < 4?
-5 -4 -3 -2 -I 1234~.X
0+0<4
-I
-2
o < 4 True

-3
-4 Since 0 is less than 4, the point (0, 0) is part of the solution set. All the points on the
-5

Isame side of the graph of x + 2y = 4 as the point (0, 0) are members of the solu-
tion set. We indicate this by shading the half-plane that contains (0, 0). The graph is
Lshown in Fig. 6.27.

I
EXAMPLE 2 Graphing an Inequality

Draw the graph of 4x - 2y ~ 12.

SOLUTION: First draw the graph of the equation 4x - 2y = 12. Use a solid line
because the points on the boundary line are included in the solution set. Now pick a

-3 -2 -I
-I
-2
I point that is not on the line. Take (0, 0) as the test point.

4x - 2y ~ 12
-3 Is 4(0) - 2(0) ~ 12?
-4
-5 •
(3, -5) o~ 12 False

-6 (0. -6)
Since 0 is not greater than or equal to 12 (0 ';t 12), the solution set is the line and
the half-plane that does not contain the point (0, 0). The graph is shown in Fig.
I 6.28.
If you had arbitrarily selected the test point (3, -5) from the other half-plane,
I you would have found that the inequality would be true: 4(3) - 2( -5) ~ 12, or
22 ~ 12. Thus, the point (3, -5) would be in the half-plane containing the solu-
tion set.

...EXAMPLE 3 Graphing an Inequality

/1 ,. Draw the graph of y < x.
5
/
4 y=x /
/ SOLUTION: The inequality is strictly "less than," so the boundary line is not part of
3
2
/
~/
/

(2,2)
I the solution set. In graphing the equation y = x, draw a dashed line (Fig. 6.29).

(O,O)~ /
Since (0, 0) is on the line, it cannot serve as a test point. Let's pick the point
(1, -1).
-5 -4 -3 -2 -,I! • 2 3 4
/ I
(I, -I)
/ -2
(-3,-3)./ -3 y<x
/
/
-4 -1 < I True
/
/.. -5
Since - 1 < 1 is true, the solution set is the half-plane containing the point
L(I,-I).
Concept/Writing Exercises
1. Outline the procedure used to graph inequalities in two
variables.
2. Explain why we use a solid line when graphing an in-
equality containing :S or :2: and we use a dashed line
when graphing an inequality containing < or>.

Practice the Skills

Tn Exercises 3-24, draw the graph of the inequality. a) Write an inequality illustrating all possible dimensions
of the rectangular garden. P = 21 + 2w is the formula
3. x :S I 4. y :2: -2 for the perimeter of a rectangle.
5. y > x +3 6. y < x - 5 b) Graph the inequality.
7. Y :2: 2x - 6 8. y < -2x +2 Challenge Problems
9. 3x - 4y > 12 10. x + 2y > 4
27. Building a House Yolanda Vega has $150,000 to spend on
11. 3x - 4y :S 9 12. 4y - 3x :2: 9 purchasing land and building a new house in the country. She
13. 3x + 2y < 6 14. -x + 2y < 2 wants at least I acre of land but less than 10 acres. If land
a costs $1500 per acre and building costs are $75 per square
15. x + y > a 16. x + 2y :S
foot, the inequality 1500x + 75y :S 150,000, where
17. 5x - 2y :S 10 18. y :2: -2x + I
I :S x < 10, describes the restriction on her purchase.
19. 3x + 2y > 12 a) What quantities do x and y represent in the inequality?
20. y :S 3x - 4 b) Graph the inequality.
c) If Yolanda decides that her house must be at least
21. ~ x - ~ Y :S I
1950 ft2 in size, how many acres of land can she buy?
22. O.lx + 0.3y :S 0.4 d) UYolanda decides that she wants to own at least 5 acres
23. 0.2x + 0.5y :S 0.3 of land, what size house can she afford?
24. ~x + ~y :2: 1 28. Men's Shirts The Tommy Hilfiger Company must ship x
men's shirts to one outlet and y men's shirts to a second
outlet. The maximum number of shirts the manufacturer
Problem Solving can produce and ship is 250. We can represent this situa-
25. Gas Grills A man-ufacturer of gas grills must produce and tion with the inequality x + y :S 250.
ship x gas grills to one outlet and y gas grills to a second a) Can x or y be negative? Explain.
outlet. The maximum number of gas grills the manufac- b) Graph the inequality.
turer can produce and ship is 300. c) Write one or two paragraphs interpreting the informa-
a) Write an inequality in two variables that represents this tion that the graph provides.
problem. 29. Which of the following inequalities have the same graph?
b) Graph the inequality. Explain how you determined your answer.
26. Flower Garden Jim Lawler has 40 ft of landscape edging a) 3x - y < 6 b) -3x + y > -6
to place around a new rectangular flower garden. c) 3x - 2y <12 d) y > 3x - 6

6.9 SOLVING QUADRATIC EQlJATIONS

BYUSING FACTORING AND BYUSING
THE QlJADRATIC FORMULA
We begin this section by discussing multiplication of binomials and factoring trinomi-
als. After we discuss factoring trinomials, we will explain how to solve quadratic
equations using factoring.
 We will now look at the FOIL method of multiplying two binomials. A binomiaL
DID YOU KNOW is an expression that contains two terms, in which each exponent that appears on a
variable is a whole number.
Aljorithuv
ecall that the series of steps I
R taken to solve a certain type of
equation is called an algorithm. Be- x +3 x - 5
cause mathematical procedures to 3x +5 4x - 2
solve equations can be generalized,
it is possible to program computers To multiply two binomials, we can use the FOIL method. The name of the
to solve problems. For example, the method, FOIL, is an acronym to help its users remember it as a method that obtains
equation 4x2 + 7x + 3 = 0 is an the products of the First, Outer, Inner, and Last terms of the binomials.
equation of the form ax2 + bx +
c = O. If you can find a procedure to L
solve an equation of the general .£.---hI First Outer Inner Last

form, it can be used to solve all I (a + b)(c + d) = a' c + a' d + b· c + b· d

equations of that form. One of the
first computer languages, FOR- I I I
TRAN (for FORmula TRANsla- o
tion), was developed specifically to
handle scientific and mathematical
applications.

EXAMPLE 1 MultipLying BinomiaLs

Multiply (x + 3)(x + 5).

F 0 I L
(x + 3)(x + 5) = x·x + x'5 + 3·x + 3'5
= x2 + 5x + 3x + 15
= x2 + 8x + 15

rEXAMPLE 2 MultipLying BinomiaLs

Multiply (2x - I )(x + 4).

F 0 I L
(2x - L) (x + 4) = 2x' x + 2x' 4 + (- I) •x + (- I).4
= 2x2 + 8x - x - 4
= 2x2 + 7x - 4

Factoring Trinomials of the Form x2 + bx + C

The expression x2 + 8x + J 5 is an example of a trinomial. A trinomiaL is an expres-
sion containing three terms in which each exponent that appears on a variable is a
whole number.
Since the product of x + 3 and x + 5 is x2 + 8x + 15, we say that x + 3 and
x + 5 are factors of x2 + 8x + 15. To factor an expression means to write the ex-
pression as a product of its factors. For example, to factor x2 + 8x + 15 we write

r 3 +5 = 8
t ,---- 3·5 = 15
x2 + 8x + 15 = (x + 3) (x + 5)

Note that the sum of the two numbers in the factors is 3 + 5 or 8. The 8 is the coeffi-
cient of the x-term. Also note that the product of the numbers in the two factors is 3 . 5,
or 15. The 15 is the constant in the trinomial. In general, when factoring an expression
of the form x2 + bx + c, we need to find two numbers whose product is C and whose
sum is b. When we determine the two numbers, the factors will be of the form

) (x + )
i i
One Other
number number

EXAMPLE 3 Factoring a Trinomial

I Factor x 2
+ 5x + 6.

I SOLUTION: We need to find two numbers whose product is 6 and whose sum is 5.
Since the product is +6, the two numbers must both be positive or both be negative.

I
Because the coefficient of the x-term is positive, only the positive factors of 6 need
to be considered. Can you explain why? We begin by listing the positive numbers
whose product is 6.

1(6) 1+6=7
(2(3) 2 +3 = 5)

Since 2 . 3 = 6 and 2 + 3 = 5, 2 and 3 are the numbers we are seeking. Thus, we

I write

I Note that (x + 3) (x + 2) is also an acceptable answer.

 1. Find two numbers whose product is c and whose sum is b.
2. Write factors in the form

(x + ) (x + )
i i
One number Other number
from step 1 from step 1

3. Check your answer by multiplying the factors using the FOIL method.

If, for example, the numbers found in step I of the above procedure were 6 and
-4, the factors would be written (x + 6) (x - 4).

rEXAMPLE 4
Factor x2 - 6x - 16.
Factoring a Trinomial

SOLUTION: We must find two numbers whose product is -16 and whose sum is
-6. Begin by listing the factors of -16.

-16(1) -16 + I = -15

(-8(2) -8 + 2 = -6)
-4( 4) -4 +4 =0
-2(8) -2 + 8=6
-1(16) -I + 16 = 15

The table lists all the factors of -16. The only factors listed whose product is -16
and whose sum is -6 are -8 and 2. We listed all factors in this example so that you
could see, for example, that -8(2) is a different set of factors than -2(8). Once
you find the factors you are looking for, there is no need to go any further. The tri-
nomial can be written in factored form as

Factoring Trinomials of the Form

ax2 + bx + c, a *" 1
Now we discuss how to factor an expression of the form ax2 + bx + c, where a, the
coefficient of the squared term, is not equal to I.
Consider the multiplication problem (2x + I)(x + 3).
(2x + 1) (x + 3) = 2x· x+ 2x· 3 + I . x + 1 . 3
= 2x2 + 6x + x + 3
= 2x + 7x + 3
2
Since (2x + l)(x + 3) = 2x2 + 7x + 3, the factors of2x2 + 7x + 3 are 2x + I
and x + 3.
Let's study the coefficients more closely.

F 0 + I L L
! ! ! .-£-hI
2x2 + 7x +3 (2x + 1)(lx + 3)
I "I I
o
o +I = (2' 3) + (1·1) = 7

Note that the product of the coefficient of the first terms in the multiplication of the
binomials equals 2, the coefficient of the squared term. The sum of the products of the
coefficients of the outer and inner terms equals 7, the coefficient of the x-term. The
product of the last terms equals 3, the constant.
A procedure to factor expressions of the form ax2 + bx + c, a *- I, follows.

To Factor Trinomial Expressions of the Form

ax2 + bx + c, a '* 1
I. Write all pairs of factors of the coefficient of the squared term, a.
2. Write all pairs of factors of the constant, c.
3. Try various combinations of these factors until the sum of the products of the
outer and inner terms is bx.
4. Check your answer by multiplying the factors using the FOIL method.

EXAMPLE 5 Factoring a Trinomial, a *' 1

I Factor 3x2 + 17x + 10.
SOLUTION: The only positive factors of 3 are 3 and 1. Therefore, we write

The number 10 has both positive and negative factors. However, since both the con-
I stant, la, and the sum of the products of the outer and inner terms, 17, are positive,
the two factors must be positive. Why? The positive factors of 10 are I (10) and
2(5). The following is a list of the possible factors.

(3x + l)(x + 10) 31x

(3x + 10)(x + 1) 13x
(3x + 2r)(x + 5) l7x ~ Correct middle term

(3x + 5)(x + 2) llx

 Note that factoring problems of this type may be checked by using the FOIL
method of multiplication. We will check the results to Example S:

(3x + 2)(x + S) = 3x· x+ 3x' S + 2· x + 2· S

= 3x2 + ISx + 2x + 10
= 3x2 + 17x + 10

. EXAMPLE 6 Factoring a Trinomial, a =1=1

Factor 6x2 - I Ix - 10.

SOLUTION: The factors of 6 will be either 6· I or 2·3. Therefore, the factors may
beoftheform(6x )(x )or(2x )(3x ).When there is more than
one set of factors for the first term, we generally try the medium-sized factors first.
If that does not work, we try the other factors. Thus, we write
6x2 - lIx - 10 = (2x )(3x

The factors of -10 are (-I )(10), (1)( -10), (-2)(S), and (2)( -S). There will
I be eight different pairs of possible factors of the trinomial 6x2 - I Ix - 10. Can
you list them?
ThecoITectfactoringis6x2 - Ilx - 10 = (2x - S)(3x + 2). A

Note that in Example 6 we first tried factors of the form (2x ) (3x ).lfwe
had not found the correct factors using them, we would have tried (6x )(x ).

Solving Quadratic Equations by Factoring

In Section 6.2, we solved linear, or first-degree, equations. In those equations, the ex-
ponent on all variables was 1. Now we deal with the quadratic equation. The stan-
dard form of a quadratic equation in one variable is shown in the box.

Note that in the standard form of a quadratic equation, the greatest exponent on x
is 2 and the right side of the equation is equal to zero. To solve a quadratic equation
means to find the value or values that make the equation true. In this section, we will
solve quadratic equations by factoring and by the quadratic formula.
To solve a quadratic equation by factoring, set one side of the equation equal to 0
and then use the zero-factor property.
 The zero-factor property indicates that, if the product of two factors is 0, then one
(or both) of the factors must have a value of O.

EXAMPLE 7 Using the Zero-Factor Property

r Solve the equation (x + 3)(x - 6) = O.

SOLUTION: When we use the zero-factor property, either (x + 3) or (x - 6) must

equal 0 for the product to equal O.Thus, we set each individual factor equal to 0
and solve each resulting equation for x.

(x + 3)(x - 6) = 0
x + 3 = 0 or x - 6 = 0
x = -3 x = 6

(x + 3)(x - 6) = 0 (x + 3)(x - 6) = 0
(-3 + 3)( -3 - 6) = 0 (6 + 3)(6 - 6) = 0
O(-9) = 0 9(0) = 0
o = 0 True o = 0 True

To Solve a Quadratic Equation by Factoring

1. Use the addition or subtraction property to make one side of the equation equal
to O.
2. Factor the side of the equation not equal to O.
3. Use the zero-factor property to solve the equation.

EXAMPLE 8 Solving a Quadratic Equation by Factoring

r Solve the equation x2 - 8x = -15.

SOLUTION: First add 15 to both sides of the equation to make the right side of the
equation equal to O.

x2 - 8x = -15
x2 - 8x + 15 = -15 + 15
x2 - 8x + 15 = 0

Factor the left side of the equation. The object is to find two numbers whose prod-
uct is 15 and whose sum is -8. Since the product of the numbers is positive and the
sum of the numbers is negative, the two numbers must both be negative. The num-
bers are -3 and -5. Note that (-3)( -5) = 15and -3 + (-5) = -8.

x2 - 8x + 15 = 0
(x - 3)(x - 5) = 0
 x-3=0 x-5=0
x = 3 x = 5

EXAMPLE 9 Solve a Quadratic Equation by Factoring

Solve the equation 3x2 - 13x +4 = O.

SOLUTION: 3x2 - 13x + 4 factors into (3x - 1)(x - 4). Thus, we write

3x2 - 13x +4 = 0
(3x - l)(x - 4) = 0
3x - 1 = 0 or x - 4 = 0
3x = 1 x = 4
1
x=-
3

TIMELY TIP As stated on page 355, every factoring problem can be checked by
multiplying the factors. If you have factored correctly, the product of the factors
will be identical to the original expression that was factored. If we wished to check
the factoring of Example 9, we would multiply (3x - I )(x - 4). Since the prod-
uct of the factors is 3x2 - 13x + 4, the expression we started with, our factoring is
correct.

Solving Quadratic Equations by Using

the Quadratic Formula
Not all quadratic equations can be solved by factoring. When a quadratic equation
cannot be easily solved by factoring, we can solve the equation with the quadratic
formula. The quadratic formula can be used to solve any quadratic equation.

Quadratic Formula
For a quadratic equation in standard form, ax2 + bx + c = 0, a * 0, the quad-
ratic formula is

In the quadratic formula, the plus or minus symbol, ±, is used. If, for example,
x = 2 ± 3, then x = 2 + 3 = 5 or x = 2 - 3 = -1.
 It is possible for a quadratic equation to have no real solution. In solving an equa-
DID YOU KNOW tion, if the radicand (the expression inside the square root) is a negative number, then
the quadratic equation has no real solution.
Tfte,M~ To use the quadratic formula, first write the quadratic equation in standard form.
oj Motwl'V Then determine the values for a (the coefficient of the squared term), b (the coeffi-
cient of the x-term), and c (the constant). Finally, substitute the values of a, b, and c
into the quadratic formula and evaluate the expression.

EXAMPLE 10 Solve a Quadratic Equation Using the Quadratic Formula

I Solve the equation x2 + 2x - 15 = 0 using the quadratic formula.

SOLUTION: In this equation, a = 1, b = 2, and c = -15.

-2 ± vl22 - 4(1)(-15)
2(1)
-2 ± V4+6O
2
-2 ± V64
he free fall of an object is some-
T thing that has interested scien-
tists and mathematicians for cen- -2 ± 8
2

turies. It is described by a quadratic 2

equation. Shown here is a time-lapse
-2 + 8 6 -2 - 8 -10
photo that shows the free fall of a ---= ~= 3 or
ball in equal-time intervals. What 2 2 2 2
you see can be described verbally
this way: The rate of change in ve-
locity in each interval is the same;
therefore, velocity is continuously Note that Example 10 can also be solved by factoring. We suggest that you do so
increasing and acceleration is con-
now.
stant.

EXAMPLE 11 Irrational Solutions to a Quadratic Equation

Solve 4x2 - 8x = -1 using the quadratic formula.

SOLUTION: Begin by writing the equation in standard form by adding 1 to both

sides of the equation, which gives the following.

-b ± vlb2 - 4ac -(-8) ± vI(-8)2 - 4(4)(1)

2a 2(4)
8 ± viM - 16
8
8±V48
8
 I Since v'48 = Vl6V3 = 4 V3 (see Section 5.4), we write
I
8±V48 8 ± 4Y3 4'(2 ± Y3) 2 ± Y3
8 8 z 2
2

2+V3 2-V3
The solutions are 2 and 2

r EXAMPLE 12 Brick Border

Diane Cecero and her husband recently installed an inground rectangular swim-
o ming pool measuring 40 ft by 30 ft. They want to add a brick border of uniform
<:')
30 -+x + width around all sides of the pool. How wide can they make the brick border if they
c':1
purchased enough brick to cover 296 ft2?

1 SOLUTION: Let's make a diagram of the pool and the brick border (Fig. 6.30) Let
x = the uniform width of the brick border. Then the total length of the larger rec-
tangular area, the pool plus the border, is 2x + 40. The total width of the larger
rectangular area is 2x + 30.
The area of the brick border can be found by subtracting the area of the pool
from the area of the pool plus the brick border.

Area of pool = ['w = (40)(30) = 1200ft2

Area of pool plus brick border = ['w = (2x + 40)(2x + 30)
= 4x2 + ]40x + ]200
Area of the brick border = area of pool plus brick border - area of pool
= (4x2 + 140x + 1200) - ] 200
= 4x2 + 140x

4x2 + 140x - 296 = 0

4(x2 + 35x - 74) = 0 Factor out 4 from each term.

4 0
- (x2 + 35x - 74) = - Divide both sides of the equation by 4.
4 4
x2 + 35x - 74 = 0
(x + 37)(x - 2) = 0 Factor trinomial.

x + 37 = 0 or x - 2 = 0
x = -37 x = 2

Since lengths are positive, the only possible answer is x = 2. Thus, they can make
a brick border 2 ft wide all around the pool.
Concept/Writing Exercises 47. x2 - 81 = 0 48. x2 - 64 = 0
1. What is a binomial? Give three examples of binomials. 49. x2+ 5x - 36 = 0 50. x 2
+ I2x + 20 = 0
2. What is a trinomial? Give three examples of trinomials. 51. 3x2 + lOx = 8 52. 3x2 - 5x = 2
3. In your own words, explain the FOIL method used to mul- 53. 5x + II x = -2
2 54. 2x2 +3= -5x
tiply two binomials. 55. 3x2 - 4x = - I 56. + I6x + 12 = 0
5x2
4. In your own words, state the zero-factor property. 57. 4x2 - 9x +2 = 0 58. 6x2 + x - 2 = 0
5. Give the standard form of a quadratic equation.
6. Have you memorized the quadratic formula? If not, you In Exercises 59-78, solve the equation, using the quadratic
need to do so. Without looking at the book, write the quad- formula. If the equation has no real solution, so state.
ratic formula.
59. x2 + 2x - 15 = 0 60. x2 + 12x + 27 = 0
61. x2 - 3x - 18 = 0 62. x2 - 6x - 16 = 0
63. x2 - 8x = 9 64. x2 = - 8x + 15
In Exercises 7-22,jactor the trinomial. If the trinomial
cannot be factored, so state. 65. x2 - 2x +3=0 66. 2x2 - x - 3 = 0
67. x2 - 4x + 2 = 0 68. 2x2 - 5x - 2 = 0
7. x2 + 9x + 18 8. x2 + 5x + 4
69. 3x 2 - 8x + I = 0 70. 2x2 + 4x + 1 = 0
9. x2 - X - 6 10. x2 + X - 6
71.4x2 - X - I = 0 72. 4x2 - 5x - 3 = 0
11. x2 + 2x - 24 12. x2 - 6x + 8
73. 2x2 + 7x + 5 = 0 74. 3x2 = 9x - 5
13. x2 - 2x - 3 14. x2 - 5x - 6
75. 3x2 - lOx + 7 = 0 76. 4x2 + 7x - I = 0
15. x2 - lOx + 2I 16. x2 - 81
77.4x2 - llx + 13 = 0 78. 5x + 9x - 2
2 = 0
17. x2 - 25 18. x2 - x - 20
19. x2 + 3x - 28 20. x2 + 4x - 32
21. x2 + 2x - 63 22. x2 - 2x - 48 Challenge Problems/Group Activities
79. Flower Garden Karen and Kurt Ohliger's backyard has a
In Exercises 23-34,jactor the trinomial. If the trinomial width of 20 meters and a length of 30 meters. Karen and
cannot be factored, so state. Kurt want to put a flower garden in the middle of the back-
yard leaving a strip of grass of uniform width around all
23. 2x2 - x - I0 24. 3x2 - 2x - 5 sides of the flower garden. If they want to have 336 square
2
25. 4x + 13x + 3 26. 2x2 - I Ix - 2 I meters of grass, what will be the width and length of the
9x + 10 garden?
27.5x + 12x + 4
2 28. 2x2 -

29. 4x2 + I Ix + 6 30. 4x2 + 20x + 21 80. Air Conditioning The yearly profit p of Arnold's Air Con-
ditioning is given by p = x2 + 15x - 100, where x is the
31. 4x2 - I I x + 6 32. 6x2 - I I x + 4 number of air conditioners produced and sold. How many
33. 3x2 - 14x - 24 34. 6x2 + 5x + I air conditioners must be produced and sold to have a
yearly profit of $45,0007
In Exercises 35-38, solve each equation, using the z,ero-
factor property.
35. (x - I)(x + 2) = 0 36. (2x + 5) (x - I) = 0
37. (3x + 4)(2x - I) = 0 38. (x - 6) (5x - 4) = 0

In Exercises 39-58, solve each equation by factoring.

39. x2 + lOx + 21 = 0 40. x2 + 4x 5 = 0
-
41. x2 - 4x +3=0 42. x2 - 5x - 24 = 0
43. x2 - 15 = 2x 44. x2 - 7x = - 6
45. x2 = 4x - 3 46. x2 - 13x + 40 = 0
81. a) Explain why solving (x - 4) (x - 7) = 6 by setting R I A A S N E Y
each factor equal to 6 is not correct. A V T R R I U P
b) Determine the correct solution to U [ I 0 N R A B
(x - 4)(x - 7) =

called the discriminant.

6.
82. The radicand in the quadratic formula, b2 - 4ac, is
How many real number solutions
will the quadratic equation have if the discriminant is
R
U
A
X
R
S
L
S
L
0
U
M
U
T
I
S
rtJ
E I I
D
C
U
N
F 0 R M M 0 N S
(a) greater than 0, (b) equal to 0, or (c) less than zero? C G E M I A L P
Explain your answer.

83. Write an equation that has solutions - I and 3. Internet/Research Activity

85. Italian mathematician Girolamo Cardano (1501-1576) is
recognized for his skill in solving equations. Write a paper
about his life and his contributions to mathematics, in par-
tiCldar his contribution to solving equations.
84. Hidden in the grid above and to the right are the following 86. Chinese mathematician Foo Ling Awong, who lived during
words discussed in this chapter: ALGEBRA, FORMULA, the Pong dynasty, developed a technique, other than trial
SOLUTION, VARIATION, BINOMIAL. You will tind and error, to factor trinomials of the form ax2 + bx + c,
these words by going letter to letter. As you move from let- a i= I. Write a paper about his life and his contributions to
ter to letter, you may move vertically or horizontally. A let- mathematics, in particular his technique for factoring tri-
ter can be used only once when spelling out each particular nomials in the form ax2 + bx + c, a i= I. (References
word. Find the words listed above in the grid. One exam- include history of mathematics books, encyclopedias, and
ple, ALGEBRA, is shown. the Internet.)

The concepts of relations and functions are extremely important in mathematics. A

relation is any set of ordered pairs. Therefore, every graph will be a relation. A func-
tion is a special type of relation. Suppose you are purchasing oranges at a supermarket
where each orange costs $0.20. Then one orange would cost $0.20, two oranges
2 X $0.20 = $0.40, three oranges $0.60, and so on. We can indicate this relation in a
table of values.

Number of Oranges Cost

0 0.00
1 0.20
2 0.40
3 0.60

10 2.00

In general, the cost for purchasing n oranges will be 20 cents times the number of or-
anges, or 0.20n. We can represent the cost, c, of n items by the equation c = 0.20n.
Since the value of c depends on the value of n, we refer to c as the dependent variable
and n as the independent variable. Note for each value of the independent variable, n,
there is one and only one value of the dependent variable, c. Such an equation is
called afunction. In the equation c = O.20n, the value of c depends on the value of n,
so we say that "c is a function of n."

A function is a special type of relation where each value of the independent vari-
able corresponds to a unique value of the dependent variable.

The set of values that can be used for the independent variable is called the
domain of the function, and the resulting set of values obtained for the dependent
variable is called the range. The domain and range for the function c = O.20n are il-
lustrated in Fig. 6.31.

Function
c = 0.20/1

When we graphed equations of the form ax + by = c in Section 6.7, we found

that they were straight lines. For example, the graph of y = 2x - 1 is illustrated in
Fig. 6.32.
Is the equation y = 2x - I a function? To answer this question, we must ask,
"Does each value of x correspond to a unique value of y?" The answer is yes; there-
fore, this equation is a function.
For the equation y = 2x - 1, we say that "y is a function of x" and write
y = f( x). The notation j(x) is read ''f of x." When we are given an equation that is a
function, we may replace the y in the equation withj(x), sincej(x) represents y. Thus,
y = 2x - 1 may be written f(x) = 2x - 1.
To evaluate a function for a specific value of x, replace each x in the function with
the given value, then evaluate. For example, to evaluate f(x) = 2x - 1 when x = 8,
we do the following.

f(x) = 2x - I
f(8) = 2(8) - 1 = 16 - 1 = 15

Thus, f(8) = 15. Since f(x) = y, when x = 8, y = 15. What is the domain and
range of f (x) = 2x - I? Because x can be any real number, the domain is the set of
real numbers, symbolized R The range is also R
We can determine whether a graph represents a function by using the vertical
line test: If a vertical line can be drawn so that it intersects the graph at more than one
point, then each x does not have a unique y and the graph does not represent a func-
tion. If a vertical line cannot be made to intersect the graph in at least two different
places, then the graph represents a function.

rEXAMPLE 1 Using the Vertical Line Test

Use the vertical line test to determine which of the graphs in Figure 6.33 represent
functions.

SOLUTION: (a), (b), and (c) represent functions, but (d) does not.
a) b) c) d)

There are many real-life applications of functions. In fact, all the applications il-
lustrated in Sections 6.2 through 6.4 are functions.
In this section, we will discuss three types of functions: linear functions, quad-
ratic functions, and exponential functions.

In Section 6.7, we graphed linear equations. The graph of any linear equation of the
form y = ax + b will pass the vertical line test, and so equations of the form
y = ax + b are linear functions. If we wished, we could write the linear function as
f( x) = ax + b since f(x) means the same as y.

EXAMPLE 2 Cost as a Linear Function

r
Adam Finiteri's weekly cost of operating a taxi, c, is given by the function
c( m) = 52 + 0.18m, where m is the number of miles driven per week. What is his
weekly cost if he drives 200 miles in a week?

SOLUTION: Substitute 200 for m in the function.

c(m) = 52 + 0.18m
c(2oo) = 52 + 0.18(200)
c(2oo) = 52 + 36 = 88
 Graphs of Linear Functions
The graphs of linear functions are straight lines that will pass the vertical line test. In
Section 6.7, we discussed how to graph linear equations. Linear functions can be
graphed by plotting points, by using intercepts, or by using the slope and y-intercept.

EXAM.PLE 3 Graphing a Linear Function

-4-3-2-1
-I Graph f(x) = -2x + 3 by using the slope and y-intercept.
-2
-3 SOLUTION: Since f(x) means the same as y, we can rewrite this function as
-4 y = -2x + 3. From Section 6.7, we know that the slope is -2 and the y-intercept
is (0, 3). Plot (0, 3) on the y-axis. Then plot the next point by moving down 2 units
and to the right 1 unit (see Fig. 6.34). A third point has been plotted in the same way.
The graph of f(x) = -2x + 3 is the line drawn through these three points.

Quadratic Functions
The standard form of a quadratic equation is y = ax2 + bx + c, a *- O. We will
learn shortly that graphs of equations of this form always pass the vertical line test
and are functions. Therefore, equations of the form y = ax2 + bx + c, a *- 0, may be
referred to as quadratic functions. We may express quadratic functions using func-
tion notation as f( x) = ax2 + bx + c. Two examples of quadratic functions are
y = 2x2 + 5x - 7 and y = -~ x2 + 4.

On July 20, 1969, Neil Armstrong became the first person to walk on the moon.
DIDYOUKNOW The velocity, v, of his spacecraft, the Eagle, in meters per second, was a function of
time before touchdown, t, given by

The height of the spacecraft, h, above the moon's surface, in meters, was also a
function of time before touchdown, given by

What was the velocity of the spacecraft and its distance from the surface of the
moon

a) v = f(t) = 3.2t + 0.45, h = get) 1.6t2 + 0.45t

f(3) = 3.2(3) + 0.45 g(3) 1.6(3)2 + 0.45(3)
A polio /I touched down at Mare
Tranqillitatis, the Sea of Tran-
quillity. The rock samples taken
=

=
9.6 + 0.45
10.05
1.6(9) + 1.35
14.4 + 1.35
there placed the age of the rocks at 15.75
3.5 billion years old, as old as the
The velocity 3 seconds before touchdown was 10.05 meters per second and the
oldest known Earth rocks.
height 3 seconds before touchdown was 15.75 meters.
 b) V = f(t) = 3.2t + 0.45, h = g(t) 1.6t2 + 0.45t
DID ;yOU KNOW f(O) = 3.2(0) + 0.45 g(O) 1.6(0)2 + 0.45(0)
= 0 + 0.45 =0+0
= 0.45 =0

The touchdown velocity was 0.45 meter per second. At touchdown, the Eagle is on
the moon, and therefore the distance from the surface of the moon is 0 meter.

Graphs of Quadratic Functions

The graph of every quadratic function is a parabola. Two parabolas are illustrated in
Fig. 6.35. Note that both graphs represent functions since they pass the vertical line
test. A parabola opens upward when the coefficient of the squared term, a, is greater
than 0, as shown in Figure 6.35(a). A parabola opens downward when the coefficient
of the squared term, a, is less than 0, as shown in Fig. 6.35(b).

tis a relatively easy matter for sci-

I entists and mathematicians to de-
scribe and predict a simple motion
y = ax2 + bx + c
a>O
y
y = ax2 + bx+ c
a<O
y
like that of a falling object. When
the phenomenon is complicated,
such as making an accurate predic-
tion of the weather, the mathematics
becomes much more difficult. The I
National Weather Service has de- /'
I
Vertex I I
vised an algorithm that takes the Axis of symmetry Axis of symmetry
temperature, pressure, moisture con-
(a) (b)
tent, and wind velocity of more than
250,000 points in Earth's atmos-
phere and applies a set of equations
that, they believe, will reasonably The vertex of a parabola is the lowest point on a parabola that opens upward and
predict what will happen at each the highest point on a parabola that opens downward. Every parabola is symmetric
point over time. Researchers at the with respect to a vertical line through its vertex. This line is called the axis of symme-
National Center for Supercomputing try of the parabola. The x-coordinate of the vertex and the equation of the axis of sym-
Applications are working on a com- metry can be found by using the following equation.
puter model that simulates thunder-
storms. Researchers want to know
why some thunderstorms can turn
severe, even deadly.
-b
x=-
2a

Once the x-coordinate of the vertex has been determined, the y-coordinate can be
found by substituting the value found for the x-coordinate into the quadratic equation
and evaluating the equation. This procedure is illustrated in Example 5.

EXAMPLE 5 Describing the Graph of a Quadratic Equation

Consider the equation y = -2x2 + 8x + 1.

r a) Determine whether the graph will be a parabola that opens upward or down-
ward.
 b) Determine the equation of the axis of symmetry of the parabola.
DID YOU KNOW c) Determine the vertex of the parabola.

SOLUTION:
a) Since a = -2, which is less than 0, the parabola opens downward.
b) To find the axis of symmetry, we use the equation x = -fa. In the equation
y = -2x2 + 8x + 1, a = -2, b = 8, and c = 1, so

-b -(8) -8
x=-=---=-=2
2a 2(-2) -4

The equation of the axis of symmetry is x = 2.

c) The x-coordinate of the vertex is 2, from part (b). To find the y-coordinate, we
substitute 2 for x in the equation and then evaluate.

y = -2x2 + 8x + 1
-2(2)2 + 8(2) + 1
= -2(4) + 16 + 1
ny golf player knows that a golf
A ball will arch in the same path
going down as it did going up. Early
=

=9
-8 + 16 + 1

gunners knew it too. To hit a distant

target, the cannon barrel was pointed
skyward, not directly at the target. A
cannonball fired at a 45° angle will
travel the greatest horizontal dis-
tance. What the golfer and gunner
alike were allowing for is the effect 1. Determine whether the parabola opens upward or downward.
of gravity on a projectile. Projectile
2. Determine the equation of the axis of symmetry.
motion follows a parabolic path.
Galileo was neither a gunner nor a 3. Determine the vertex of the parabola.
golfer, but he gave us the formula 4. Determine the y-intercept by substituting x = 0 into the equation.
that effectively describes that motion 5. Determine the x-intercepts (if they exist) by substituting y = 0 into the equation
and the distance traveled by an ob- and solving for x.
ject if it is projected at a specific an-
6. Draw the graph, making use of the information gained in steps 1 through 5. Re-
gle with a specific initial velocity.
member the parabola will be symmetric with respect to the axis of symmetry.

In step 5, to determine the x-intercepts, you may use either factoring or the quad-
ratic formula.

EXAMPLE 6 Graphing a Quadratic Equation

Sketch the graph of the equation y = x2 - 6x + 8.

SOLUTION: We follow the steps outlined in the general procedure.

1. Since a = 1, which is greater than 0, the parabola opens upward.
-b -(-6) 6
2. Axis of symmetry: x = - = --- = - = 3
2a 2(1) 2
Thus, the axis of symmetry is x = 3.
3. y-coordinate of vertex: y = x2 - 6x + 8
y = (3)2 - 6(3) + 8 = 9 - 18 + 8 = -1
Thus, the vertex is at (3, - 1).
 4. y-intercept: y = x2 - 6x + 8
y = 02 - 6(0) + 8 = 8
Thus, the y-intercept is at (0, 8).
5. x-intercepts: 0 = x2 - 6x + 8, or x2 - 6x + 8 = 0
We can solve this equation by factoring.

x2 - 6x + 8 = 0
(x - 4)(x - 2) = 0
x - 4 = 0 or x - 2 = 0
x=4 x=2

Thus, the x-intercepts are (4, 0) and (2,0).

6. Plot the vertex (3, -1), the y-intercept (0,8), and the x-intercepts (4, 0) and
(2, 0). Then sketch the graph (Fig. 6.36). •.

Note that the domain of the graph in Example 6, the possible x-values, is the set
of all real numbers, IR.The range, the possible y-values, is the set of all real numbers
greater than or equal to -1. When graphing parabolas, if you feel that you need addi-
tional points to graph the equation, you can always substitute values for x and find the
corresponding values of y and plot those points. For example, if you substituted 1 for
x, the corresponding value of y is 3. Thus, you could plot the point (l, 3).

EXAMPLE 7 Domain and Range of a Quadratic Function

= - 2x2 + 3x + 4.
a) Sketch the graph of the function f( x)
b) Determine the domain and range of the function.

SOLUTION:
a) Since fix) means y, we can replace fix) with y to obtain y = -2x2 + 3x + 4. Now
graph y = -2x2 + 3x + 4 using the steps outlined in the general procedure.

1. Since a = -2, which is less than 0, the parabola opens downward.

-b -(3) -3 3
2. Axis of symmetry: x = - = --- = - = -
2a 2(-2) -4 4

Thus, the axis of symmetry is x = ~.

4

-2(~) 16
+ 2. +
4
4
9 9
-- + - + 4
8 4
9 18 32 41 1
-- + - +- = - or 5-
8 8 8 8 8
 Thus, the y-intercept is (0, 4).

5. x-intercepts: y = -2x2 + 3x + 4

This equation cannot be factored, so we will use the quadratic formula to

solve it.

a = -2, b = 3,

-b ± Vb2 - 4ac
2a
-3 ± V32 - 4(-2)(4)
2( -2)
-----.0.
\4'
51)8
-3 ± v9+32
!(x) = - 2x2 + 3x + 4
-4
-3 ± v4T
-4

x ~ _-_3_+_6_.4~ _3._4~ -0.85 -3 - 6.4 -9.4

x ~ --- ~ -- ~ 2.35
-4 -4 -4 -4
Thus, the x-intercepts are (-0.85,0) and (2.35, 0).

6. Plot the vertex (t 5 i), the y-intercept (0, 4), and the x-intercepts (-0.85,0)
and (2.35, 0). Then sketch the graph (Fig. 6.37).
The domain, the values that can be used for x, is the set of all real numbers, III
The range, the values of y, is y ::; 5 i.
When we use the quadratic formula to find the x-intercepts of a graph, if the radi-
cand, b2 -:- 4ac, is a negative number, the graph has no x-intercepts. The graph will lie
totally above or below the x-axis.

Exponential Functions
In Section 6.3, we discussed exponential equations. Recall that exponential equations
are of the form y = aX, a > 0, a =F- I. The graph of every exponential equation will
pass the vertical line test, and so every exponential equation is also an exponential
function. Exponential functions may be written as j(x) = aX, a > 0, a =F- 1.
In Section 6.3, we also introduced the natural exponential formula P = Poekt. We
can write this formula in function notation as P(t) = Poekt. This expression is re-
ferred to as the natural exponential junction. In Example 8, we use the natural expo-
nential function.
rEXAMPLE 8 Evaluating an Exponential Decay Function
The power supply of a satellite is a radioisotope. The power output, p, in watts re-
maining in the power supply is a function of the time the satellite is in space. If
there are originally 100 grams of the radioisotope, the power remaining after t days
is p(t) = 100e-O.OOlt. What will be the remaining power after 1 year (or 365 days)
in space?

SOLUTION: Substitute 365 days for t in the function, and then evaluate using a cal-
culator as described in Section 6.3.

p(t) 100e-O.OOlt
p(365) 100e-O.OOI (365)
100e-O.365
;:,j 100(0.6941966509)
;:,j 69.4 watts

rEXAMPLE 9 Evaluating an Exponential Decay Function

Carbon 14 is used by scientists to find the age of fossils and other artifacts. If an
object originally had 25 grams of carbon 14, the amount present after t years is
f(t) = 25e-o.OOOI2010t. How much carbon 14 will be found after 350 years?

SOLUTION: Substitute 350 for t in the function, and then evaluate using a calculator
as described in Section 6.3.

fit) = 25e-00001201Ot
f(350) = 25e-o.OOOI2010(350)
= 25e-O.042035
;:,j 25(0.9588362207)
;:,j 23.97090552
;:,j 24 grams

Graphs of Exponential Functions

What does the graph of an exponential function of the form y = aX, a > 0, a * 1,
look like? Examples 10 and II illustrate graphs of exponential functions.

EXAMPLE 10 Graphing an Exponential Function, a > 1

ra) Graph y = 2X•

b) Determine the domain and range ofthe function.
 SOLUTION:
DID YOU KNOW a) Substitute values for x and find the corresponding values of y. The graph is
shown in Fig. 6.38.

y = 2X
opulation growth during certain
P time periods can be described by
an exponential function. Whether it x= -3, y = 2-3 =
1
=
1
-
x y

1
23 8 -3 -
is a population of bacteria, fish, 8
flowers, or people, the same general 1 1
trend emerges: a period of rapid (ex-
x= -2, Y -- r2 -
- -22 --
- 4 1
-2 -
ponential) growth, which is then fol- 4
lowed by a leveling-off period. 1 1
x= -1, Y = r l =
21
-
2 -1
1
-
2
x = 0, y = 2° = 1 0 1
x= 1, Y = 21 =2 1 2
x = 2, y = 22 = 4 2 4
x = 3, y = 23 = 8 3 8

b) The domain is all real numbers, IR. The range is y > O. Note that y can never
have a value of O. •••

All exponential functions of the form y = aX, a > 1, will have the general shape
of the graph illustrated in Fig. 6.38. Since f(x) is the same as y, the graphs of functions
of the form f(x) = aX, a > 1, will also have the general shape of the graph illustrated
in Fig. 6.38. Can you now predict the shape of the graph of y = eX? Remember: e has
a value of about 2.7183.

EXAMPLE 11 Graphing an Exponential Function, 0 < a <

a) Graph y = (~Y.
b) Determine the domain and range of the function.

SOLUTION:·
a) We begin by substituting values for x and calculating values for y. We then plot
the ordered pairs and use these points to sketch the graph. To evaluate a fraction
with a negative exponent, we use the fact that
 Then
DID YOU KNOW
y= (~y x y

t3
-3 8
-2 4
3
x = -3, y= (1"2 = 2 = 8
-1 2

x = -2, y= (~t2 = 22 = 4
0 1
1
-
2
x = -1, y= (1- )-' = 2' =2 1
2 2 -
4
x = 0, y= (~y 3
1
-
8

x = 1, y= (~y 1
-
2

x = 2, y=
(~y 1
-
4
E lectronic mail, or e-mail, is fast
becoming one of the most popu-
lar and easiest methods of communi-
cating. The number of e-mail mes-
x = 3, y= (~y 1
-
8
sages is increasing exponentially.
Although 45% of the U.S. popula- The graph is illustrated in Fig. 6.39.
tion now uses e-mail, according to
Jupiter Research, 93% of people on- y
line frequently use e-maiI.Re-
searchers expect this percentage to
increase to 98% by 2007. Nearly
one-quarter of e-mail users maintain
three or more personal e-mail ac-
counts. Thirty-five percent of e-mail
messages received is considered
"spam mail" or unsolicited e-mail,
whereas e-mail from friends and
family makes up 34% of messages
received. E-mail is also becoming a
valuable marketing tool. According
to the Kelsey Group, annual spend-
ing by U.S. small business on local-
ized e-mail marketing will exceed
$2.2 billion by 2005.

All exponential functions of the form y = aX or f(x) = aX, 0 < a < 1, will
have the general shape of the graph illustrated in Fig. 6.39.

EXAMPLE 12 Is the Growth Exponential?

Sales of Botox, a drug injected under the skin to smooth out wrinkles, have in-
creased tremendously since the late 1990s. The graph on page 372 shows the sales
of Botox from 1999 through 2001 and projected though 2005.
 a) Does the graph approximate the graph of an exponential function?
b) Estimate the sales of Botox in 2004.

SOLUTION:
a) Yes, the graph has the approximate shape of an exponential function. A function
that increases rapidly with this general shape, is, or approximates, an exponen-
tial function.
b) From the graph, we see that in 2004, sales of Botox were about $800 million.

1100
1000
900
V> 800
S 700
·s 600
~ 500
~ 400
0;
U) 300
200
100
o
1999 2001 2003
Year
Source: American Society of Plastic
Surgeons, Allergan

'7~ SECTION 6.10 EXERCISES .:>:~~

..
Concept/Writing Exercises 9. 10.
1. What is a function?
2. What is a relation?
3. What is the domain of a function? -2 x
4. What is the range of a function?
5. Explain how and why the vertical line test can be used to
determine whether a graph is a function.
6. Give three examples of one quantity being a function of 11. y 12. y
another quantity.
2

Practice the Skills

-2 2 x -2 2 x
In Exercises 7-24, determine whether the graph represents
-2 2
a function. If it does represent a function, give its domain
and range.
7. y 8. y 13. 14. y
• 15
2
•• • .2

• •
-2 2 x -2 x 15 x

-2 -2
2
•
• -6
 6.10 Functions and Their Graphs 373

15. y 16. y In Exercises 25-30, determine whether the set of ordered

1 pairs is a function.

25. {(2, 9), (3,6), (7, 11), (9, IS)}

x -2 x 26. {(I, -3), (2, -5), (3, -7), (4, -9)}
27. {(4, 4), (5, 1), (4, O)}
-I 28. {( 3, 1), (3, 3), (3, 5)}
29. {(7, 1), (6, 1), (5, I)}
17. y 18. 30. {( 1, 7), (1, 6), (1, 5)}
....-0
2 ....-0 In Exercises 31-44, evaluate the function for the given
value ofx.
x x
-8 31. f(x) = x + 3, x = 2
-2 32. f(x) = 2x + 5, x = 4
33. f(x) = -2x - 7, x = -4
34. f(x) = -5x + 3, x = -1
19. y 20. 35. f(x)
36. f(x)
=
=
lOx - 6,
7x - 6,
x =
x=4
°
37. f(x) = x2 - 3x + 1, x = 4
38. f(x) = x2 - 5, x=7
-2 2 x 30 x
39. f(x) = 2x2 - 2x - 8, x = -2
40. f(x) = -x2 + 3x + 7, x=2
41. f(x) = -3x2 + 5x + 4, x = -3
42. f(x) = 5x + 2x + 5, x=4
2
21. 22. y
43. f(x) = -5x2 + 3x - 9, x = -1
44. f(x) = -3x2 - 6x + 10, x = -2
2

x In Exercises 45-50, graph the function by using the slope

-4 4
and y-intercept.
-4
-2 45. f(x) = 2x - 1 46. f (x) = -x +3
47. f(x) = -4x + 2 48. f (x) = 2x + 5
23. y
49.f(x) = ~x - 1 50. f(x) = -~x + 3
4
2 In Exercises 51-66,
a) determine whether the parabola will open upward or
downward.
-4 -2 2 4 x
b) find the equation of the axis of symmetry.
-2 c) find the vertex.
-4 d) find the y-intercept.
e) find the x-intercepts if they exist.
f) sketch the graph.
24. y g) find the domain and range of the function.

51. y = x2 - 16 52. Y = x2 - 9
53. y = -x2 + 4 54. Y = -x2 + 16
-4 -2 2 56. y = -2x2 - 8
55. f(x) = -x2 - 4
-2
57. Y = 2x2 - 3 58. f(x) = -3x2 - 6
59. f(x) = x2 + 2x + 6 60. y = x2 - 8x + 1
61. y = x2 + 5x + 6 62. y = X 2 - 7x - 8 c) Determine the x-coordinate of the vertex, then use this
+ 8x - 8 value in the function f(x) to estimate the minimum per-
63. y = - x2 + 4x - 6 64. y = - x2
centage of fust-time California State University fresh-
65.y = -3x2 + 14x - 8 66. y = 2x2 - X - 6 men who entered college with college-level mathemat-
ics proficiency.
In Exercises 67-78, draw the graph of the function and Percentage of First·time California State University Freshmen
state the domain and range. Entering with College-level Mathematics Proficiency

67. y = 3X 68. f(x) = 4x

69. y = (~y 70. Y = (±y

71.f(x) = 2 X + 1 72. y = 3x - 1
73. y = 4x + 1 74. y = 2x - 1 C 60
75. y = 3x-1 76. Y = 3x+1 '2"
&;
77. f(x) = 4x+1 78. y = 4x-1 40

Problem Solving
79. Monthly Salary Chet Rogalski is part owner of a newly 0'---'---'---'---
opened hardware store. Chefs monthly salary is given by ~ ~ ~ ~ ~ ~ ~ ~ m
thefunctionm(s) = 300 + O.IOs,wheresisthestore's Year
monthly sales in dollars. If sales for the month of July are Source: California State University Board of Trustees report
$20,000, determine Chet's monthly salary for July. 82. Free Meals The following graph indicates the number of
80. Finding Distances The distance a car travels, d(t), at a free lunches, in thousands, served in the Rochester, NY,
constant 60 mph is given by the function d (t) = 60t, Summer Meals Program from 1994 through 2000. The
where t is the time in hours. Find the distance traveled in function l(x) = -4.25x2 + 30.32x + 150.14 can be
used to estimate the number of free lunches served, lex),
where x is the number of years since 1994 and
o :S x :S 6.
a) Use the function lex) to estimate the number of free
lunches served in 1999.
b) Use the graph to determine the year in which the num-
ber of free lunches served was a maximum.
c) Determine the x-coordinate of the vertex, then use this
value in the function lex) to estimate the maximum
number of free lunches served.
a) 3 hours.
b) 7 hours.
81. The following graph indicates the percent of first-time
California State University freshmen who entered college
with college-level mathematics proficiency for the years
200
1992 through 2001. The function f(x) = 0.56x2 - 'Vi
"0
5.43x + 59.83 can be used to estimate the percent of c:
;J.
first-time California State University freshmen who en- .s0" 150
tered college with college-level mathematics proficiency, ~
fix), where x is the number of years since 1992 and "
'>"
o :S x :S 9. ~ 100
a) Use the functionf(x) to estimate the percent of first- '"
time California State University freshmen who entered '"
;;s'"
college with college-level mathematics proficiency in 50
2000.
b) Use the graph to determine the year in which the per-
0
cent of fust-time California State University freshmen '94 '95 '96 '97 '98 '99 '00
who entered college with college-level mathematics Year
proficiency was a minimum. Source: YWCA
83. Expected Growth The town of Lockport currently has a) From 1995 through 2001 projected through 2003, does
4000 residents. The expected future population can be ap- the graph approximate the graph of an exponential
proximated by the function P(x) = 4000( 1.3 )O.lx, where function? Explain.
x is the number of years in the future. Find the expected b) Estimate the U.S. average cost of a PC in 2002.
population of Lockport in 87. The spacing of the frets on the neck of a classical guitar is
a) 10 years. determined from the equation d = (21.9)(2)(20-x)/I2,
b) 50 years. where x = the fret number and d = the distance in cen-
84. Decay of Plutonium Plutonium, a radioactive material timeters of the xth fret from the bridge.
used in most nuclear reactors, decays exponentially at a
rate of 0.003% per year. The amount of plutonium, P, left
after t years can be found by the formula
P = Poe-O.00003!, where Po is the original amount of plu-
tonium present. If there are originally 2000 grams of pluto-
nium, find the amount of plutonium left after 50 years.
85. Scooter Injuries The number of scooter injuries rose I kut
rapidly during the summer months in 2000. The graph I 1st fret
below shows the number of scooter injuries, by month, 2nd fret
in 2000.
a) Does the graph approximate the graph of an exponen-
tial function from May through September 2000? a) Determine how far the 19th fret should be from the
b) Estimate the number of scooter injuries in August bridge (rounded to one decimal place).
2000. b) Determine how far the 4th fret should be from the
bridge (rounded to one decimal place).
c) The distance of the nut from the bridge can be found by
letting x = 0 in the given exponential equation. Find
the distance from the nut to the bridge (rounded to one
decimal place).

Challenge Problems/Group Activities

88. Appreciation of a House A house initially cost $85,000.
The value, V, of the house after n years if it appreciates at
a constant rate of 4% per year can be determined by the
function V = f( n) = $85,000( 1.04 )n.
a) Determine f(8) and explain its meaning.
o b) After how many years is the value of the house greater
Jan. Feb. Mar. Apr. May. Jun. Jul. Aug. Sep. Oct. ov. Dec.
than $153,000? (Find by trial and error.)
Source: Consumer Product Safety Commission
89. Target Heart Rate While exercising, a person's recom-
86. Cost of a PC The graph shows the U.S. average cost of a mended target heart rate is a function of age. The recom-
personal computer (PC), in thousands of dollars, from mended number of beats per minute, y, is given by the
1995 through 2001 projected through 2003. functiony = f(x) = -0.85x + 187, where x represents
a person's age in years. Determine the number of recom-
U.S. Average PC Cost mended heart beats per minute for the following ages and
In thousands of dollars explain the results.
a) 20
$2.5~ b) 30
c) 50
2.0~
d) 60
1.5 ~ e) What is the age of a person with a recommended target
1.0 ---------- heart rate of 85?
0.5 ---------- 90. Speed of Light Light travels at about 186,000 miles per
I I I I second through space. The distance, d, in miles that light
1995 97 99 2001
travels in t seconds can be determined by the function
d(t) = 186,000t.
 a) Light reaches the moon from Earth in about 1.3 sec.
Determine the approximate distance from Earth to the
moon. 91. The idea of using variables in algebraic equations was in-
b) Express the distance in miles, d, traveled by light in t troduced by the French mathematician Franc;ois Viete
minutes as a function of time, t. (1540-1603). Write a paper about his life and his contribu-
c) Light travels from the sun to Earth in about 8.3 min. tions to mathematics. In particular, discuss his work with
Determine the approximate distance from the sun to algebra equations. (References include history of mathe-
Earth. matics books, encyclopedias, and the Internet.)

Z~h
CHAPTER 6

Properties used to solve equations Y2 - Yl

Slope (m): m = ---
X2 - Xl
Addition property of equality
If a = b, then a + e = b +e Equations and formulas
Subtraction property of equality Linear equation in two variables:
If a = b, then a - e = b - e ax + by = e, a =I' 0, b =I' 0
Multiplication property of equality Quadratic equation in one variable:
If a = b, then ae = be
ax2 + bx + e = 0, a =I' 0
Division property of equality
If a = b, then ale = ble, e =I' 0 Quadratic equation (or function) in two variables:
Variation y = ax2 + bx + e, a =I' 0
Direct: y = kx Exponential equation (or function):
k y = aX, a =I' 1, a > 0
Inverse: y = -x
Exponential growth or decay formula:
Joint: y = kxz P = Poakt, a =I' 1, a >0
Inequality symbols
Quadratic formula:
a < b means that a is less than b. -b ± \lb2 - 4ae
a :5 b means that a is less than or equal to b.
2a
a > b means that a is greater then b. Slope-intercept form of a line:
a 2': b means that a is greater than or equal to b. y=mx+b
Intercepts Axis of symmetry of a parabola:
To find the x-intercept, set y = 0 and solve the resulting -b
x=-
equation for x. 2a
To find the y-intercept, set x = 0 and solve the resulting Zero-factor property
equation for y. If a . b = 0, then a = 0 or b = O.

6.1 3. 4x2 - 2x + 5,
In Exercises 1-6, evaluate the expressionjor the given
4. -x2 + 7x - 3,
valuer s) oj the variable.
1. x2 + 12, x = 3
S. 4x3 - 7x2 + 3x + 1, x = -2

2. -x2 - 9, x = -1 6. 3x2 - xy + 2y2, X = 1, y = -2

 30. 3 times y decreased by 7
31. 10 increased by 3 times r
32. The difference between 8 divided by q and 11
7. 3x - 4 + x +5 8.3x + 4(x - 2) + 6x
9. 4(x - 1) + ~(9x + 3) In Exercises 33-36, write an equation that can be used to
solve the problems. Solve the equation and find the desired
valuers).
10.4s + 10 = -30
x+5 x-3 33. Four increased by 3 times a number is 22.
12.----- =----- 34. The product of 3 and a number, increased by 8, is 6 less
6 3
than the number.
x 3
14. - +- = 7 35. Five times the difference of a number and 4 is 45.
4 5
36. Fourteen more than 10 times a number is 8 times the sum
15. Making Oatmeal A recipe for Hot Oats Cereal calls for 2
of the number and 12.
cups of water and for ~ cup of dry oats. How many cups of
dry oats would be used with 3 cups of water?
In Exercises 37-40, write the equation and then find the
16. Laying Blocks A mason lays 120 blocks in 1 hr 40 min.
solution.
How long will it take her to lay 300 blocks?
37. MODELING· Investing Jim Lawton received an inheritance
of $15,000. If he wants to invest twice as much money in
In Exercises 17-20, use the formula to find the value of the mutual funds as in bonds, how much should he invest in
indicated variable for the values given. mutual funds?
38. MODELING· Lawn Chairs Larry's Lawn Chair Company
17. A = bh
has fixed costs of $15,000 per month and variable costs of
Find A when b = 12 and h = 4 (geometry). $9.50 per lawn chair manufactured. The company has
18. V = 21T R2r2 $95,000 available to meet its total monthly expenditures.
Find V when R = 3, r = 1~, and 1T = 3.14 What is the maximum number of lawn chairs the company
(geometry). can manufacture in a month? (Fixed costs, such as rent and
insurance, are those that occur regardless of the level of
x-J.L
19.z =-- production. Variable costs, such as those for materials, de-
a pend on the level of production.)
vn 39. MODELING· ZOO Animals According to the 2003 World
Find x when z = 2, J.L = 100, a = 3, and n = 16 Almanac, the number of species at the San Diego Zoo is
(statistics). 140 more than two times the number of species at the
20. K = ~mv2 Philadelphia Zoo. The sum of the species at the San Diego
Zoo and the Philadelphia Zoo is 1130. Determine the num-
Find m when v = 30 and k = 4500 (physics).
ber of species at each zoo.

21. 3x - 9y = 18 22. 2x + 5y = 12
23. 2x - 3y + 52 = 30 24. -3x - 4y + 5z = 4

25. A = lw, for w

26. P = 21 + 2w, for w
27. L = 2(wh + lh), for I
28. an = a I + (n - 1 )d, for d
40. MODELING· Restaurant Profit John Smith owns two
restaurants. His profit for a year at restaurant A is $12,000
greater than his profit at restaurant B. The total profit from
both restaurants is $68,000: Determine the profit at each
restaurant.
In Exercises 41-44, find the quantity indicated. 63. x - y = 4 64. 2x + 3y = 12

41. s is inversely proportional to t. If s = 10 when t = 3, find 65. x = y 66. x = 3

s when t = 5.
In Exercises 67-70, graph the equation, using the x- and y-
42. J is directly proportional to the square of A. If J = 32
intercepts.
when A = 4, find J when A = 7.
43. W is directly proportional to L and inversely proportional 67. x - 2y = 6 68. x + 3y = 6
toA. IfW = 80 when L = 100 and A = 20, find W 69.4x - 3y = 12 70. 2x + 3y = 9
when L = 50 and A = 40.
44. z is jointly proportional to x and y and inversely propor- In Exercises 71-74, find the slope of the line through the
tional to the square of r. If z = 12 when x = 20, y = 8, given points.
and r = 8, find z when x = 10, y = 80, and r = 3.
45. Buying Fertilizer 71. (1,3), (6, 5) 72. (3, -1), (5, -4)
a) A 30 Ib bag of fertilizer will cover an area of 2500 ft2. 73. (-1, -4), (2,3) 74. (6,2), (6, -2)
How many pounds of fertilizer are needed to cover an
area of 12,500 ft2? In Exercises 75-78, graph the equation by plotting the
b) How many bags of fertilizer are needed? y-intercept and then plotting a second point by making
46. Map Reading The scale of a map is 1 in. to 30 mi. What use of the slope.
distance on the map represents 120 mi?
47. Electric Bill An electric company charges $0.162 per kilo-
75.y = 2x - 5 76. 2y - 4 = 3x
watt-hour (kWh). What is the electric bill if 740 kWh are 77. 2y + x = 8 78. y = -x - 1
used in a month?
48. A Falling Object The distance, d, an object drops in free fall
is directly proportional to the square of the time, t. If an ob-
ject falls 16 ft in 1 sec, how far will an object fall in 5 sec?

In Exercises 49-52, graph the solution set for the set of real
numbers.
49.5 + 9x ~ 7x - 7 50. 2x + 8 2:: 4x + 10
51. 3(x + 9) ~ 4x + 11 52. - 3 ~ x + 1< 7
81. Disability Income The monthly disability income, I, that
Nadja Muhidin receives is I = 460 - O.5m, where m is
In Exercises 53-56, graph the solution setfor the set of
her monthly earnings for her part-time job for the previous
integers. month.
53.2 + 5x > -8 54.5x + 13 2:: -22 a) Draw a graph of disability income versus earnings for
55. -1 < x ~ 9 56. - 8 ~ x +2 ~ 7 earnings up to and including $920.
b) If Nadja earns $600 in January, how much disability in-
come will she receive in February?
c) If she received $380 disability income in November,
In Exercises 57-60, graph the ordered pair in the Cartesian how much did she earn in October?
coordinate system. 82. Business Space Rental The monthly rental cost, C, in dol-
lars, for space in the Galleria Mall can be approximated by
the equation C = 1.70A + 3000, where A is the area, in
In Exercises 61 and 62, points A, B, and C are vertices of a square feet, of space rented.
rectangle. Plot the points. Find the coordinates of the a) Draw a graph of monthly rental cost versus square feet
fourth point, D, to complete the rectangle. Find the area of for up to and including 12,000 ft2.
the rectangle. b) Determine the monthly rental cost if 2000 ft2 are
rented.
61. A(-3,3), B(2,3), C(2, -1) c) If the rental cost is $10,000 per month, how many
62. A( -3, I), B(-3, -2), C(4, -2) square feet are rented?
 105.f(x) = 5x - 2, x = 4
83.4x + 3y:s 12 84. 3x + 2y ~ 12 106. f(x) = -2x + 7, x = -3
85. 2x - 3y > 12 86. -7 x - 2Y < 14 107. f(x) = 2x2 - 3x + 4, x = 5
108.f(x) = -4x2 + 7x + 9, x = 4

In Exercises 87-92, factor the trinomial. If the trinomial In Exercises 109 and 110, for each function
cannot be factored, so state. a) determine whether the parabola will open upward or
downward.
87. x2 + 9x + 18 88. x2 + x - 20 b) find the equation of the axis of symmetry.
89. x2 - lOx + 24 90. - 9x + 20
x2 c) find the vertex.
91. 6x2 + 7x - 3 92. 2x2 + 13x - 7 d) find the y-intercept.
e) find the x-intercepts if they exist.
In Exercises 93-96, solve the equation by factoring. f) sketch the graph.
g) find the domain and range.
93. x2 + 3x + 2 = 0 94. x2 - 5x = -4
95. 3x - 17x + 10 = 0
2 96. 3x2 = -7 x - 2 109.y = -x2 - 4x + 21
In Exercises 97-100, solve the equation, using the quad- 110.f(x) = 2x2 - 8x+ 10
ratic formula. If the equation has no real solution, so state.
In Exercises 111 and 112, draw the graph of the function
97. x2 - 4x - 1 = 0 98. x2 - 3x +2 = 0 and state the domain and range.
99. 2x2 - 3x +4 = 0 100. 2x2 - X - 3 = 0

In Exercises 101-104, determine whether the graph repre-

sents a function. If it does represent a function, give its do- 113. Gas Mileage The gas mileage, m, of a specific car can be
main and range. estimated by the equation (or function)
101. y 102. m = 30 - 0.002n2, 20 :s n :s 80
3
• where n is the speed of the car in miles per hour. Estimate
the gas mileage when the car travels at 60 mph.
-3 3 x -6 114. Auto Accidents The approximate number of accidents in
• • one month, n, involving drivers between 16 and 30 years
of age inclusive can be approximated by the equation
-3

103. y 104. where a is the age of the driver. Approximate the number
3 of accidents in one month that involved
a) 18-year-olds
b) 25-year-olds.
115. Filtered Light The percent of light filtering through Swan
Lake, P, can be approximated by the function
P(x) = 100(0.92Y, where x is the depth in feet. Find
the percent of light filtering through at a depth of 4.5 ft.
 ~

CHAPTER 6 TEST

12. y = 2x - 4 13. 2x - 3y = 15
14. Graph the inequality 3y 2 5x - 12.
2. 3x +5 = 2(4x - 7) 15. Solve the equation x2 - 3x = 28 by factoring.
3. - 2 (x - 3) + 6x = 2x + 3 (x - 4) 16. Solve the equation 3x2 + 2x = 8 by using the quad-
ratic formula.
Tn Exercises 4 and 5, write an equation to represent the 17. Determine whether the graph is a function. Explain
problem. Then solve the equation. your answer.
4. The product of a number and 2, increased by 7 is 25.
5. Buying a Car The cost of a car including a 7% sales
tax is $26,750. Determine the cost of the car before
tax.
6. Evaluate L = ah + bh + ch when
a = 3, b = 4, c = 5, and h = 7.
7. Solve3x + 5y = 11 fory.
8. L varies jointly as M and N and inversely as P. If
L = 12 when M = 8, N = 3, and P = 2, find L
18. Evaluate f(x) = -4x2 - llx + 5 when x = -2.
when M = 10, N = 5, and P = 15. 19. For the equation y = x2 - 2x + 4,
a) determine whether the parabola will open upward
9. For a constant area, the length, I, of a rectangle varies
or downward.
inversely as the width, w. If 1 = 15 ft when
b) find the equation of the axis of symmetry.
w = 9 ft, find the length of a rectangle with the same
c) find the vertex.
area if the width is 20 ft.
d) find the y-intercept.
10. Graph the solution set of - 3x + 11 :s: 5x + 35 on e) find the x-intercepts if they exist.
the real number line. f) sketch the graph.
11. Determine the slope of the line through the points g) find the domain and range of the function.
(-3,5) and (7,12).
GROUP PROJECTS
length of his or her long bones, decreases at the rate
of 0.06 cm per year after the age of 30.
1. Archeologists have developed formulas to predict the i. At age 30, Jolene is 168 cm tall. Estimate the
height and, in some cases, the age at death of the de- length of her humerus.
ceased by knowing the lengths of certain bones in the ii. Estimate the length of Jo]ene's humerus when
body. The long bones of the body grow at approxi- she is 60 years old.
mately the same rate. Thus, a linear relationship exists f) Select six people of the same gender and measure
between the length of the bones and the person's their height and one of the bones for which an
height. If the length of one of these major bones-the equation is given (the same bone on each person).
femur (F), the tibia (T), the humerus (H), and the radius Each measurement should be made to the nearest
(R)- is known, the height, h, of a person can be calcu- 0.5 cm. For each person, you will have two meas-
lated with one of the following formulas. The relation- urements, which can be considered an ordered pair
ship between bone length and height is different for (bone length, height). Plot the ordered pairs on a
males and females. piece of graph paper, with the bone length on the
Male Female horizontal axis and the height on the vertical axis.
h = 2.24F + 69.09 h = 2.23F + 61.41 Start the scale on both axes at zero. Draw a straight
h = 2.39T + 81.68 h = 2.53T + 72.57 line that you fee] is the best approximation, or best
fit, through these points. Determine where the line
h = 2.97H + 73.57 h = 3.14H + 64.98
crosses the y-axis and the slope of the line. Your y-
h = 3.65R + 80.41 h = 3.88R + 73.51 intercept and slope should be close to the values in
the given equation for that bone. (Reference: M.
Trotter and G. C. Gieser, "Estimation of Stature
from Long Bones of American Whites and Ne-
groes," American Journal of Physical Anthropol-
ogy, 1952, ]0:463-514.)

2. The functions that we graphed in this chapter can be

easily graphed with a graphing calculator (or grapher).
If you do not have a graphing calculator, borrow one
from your instructor or a friend.
a) Exp]ain how you would set the domain and range.
a) Measure your humerus and use the appropriate for- The calculator key to set the domain and range may
mula to predict your height in centimeters. How be labeled range or window. Set the grapher with
close is this predicted height to your actual height? the following range or window settings:
(The result is an approximation because measuring Xmin = -12, Xmax = ]2, Xscl = 1,
a bone covered with flesh and muscle is difficult.) Ymin = -13, Ymax = 6, and Yscl = 1.
b) Determine and describe where the femur and tibia b) Exp]ain how to enter a function in the graphing ca]-
bones are located. cu]ator. Enter the function y = 3x2 - 7x - 8 in
c) Dr. Juarez, an archeologist, had one female the calculator.
humerus that was 29.42 cm in length. He concluded c) Graph the function you entered in part (b).
that the height of the entire skeleton would have d) Learn how to use the trace feature. Then use it to
been 157.36 cm. Was his conclusion correct? estimate the x-intercepts. Record the estimated val-
d) If a 21-year-old woman is 167.64 cm tall, about ues for the x-intercepts.
how long should her tibia be? e) Learn how to use the zoom feature to obtain a better
e) Sometimes the age of a person may be determined approximation of the x-intercepts. Use the zoom
by using the fact that the height of a person, and the feature twice and record the x-intercepts each time.