ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

MATH 1010- Algebra & Trigonometry

Summer 2024

Unit 1
Linear equations and linear inequalities

Author: Ali Saad

Department of Mathematics
College of Education
University of Doha for Science and Technology

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

Section 1.1 Linear equations in one variable

Definition A linear equation in one variable has the form: 𝑎𝑥 + 𝑏 = 𝑐 , where
𝑎, 𝑏 , 𝑎𝑛𝑑 𝑐 are real numbers and 𝑎 is not equal to zero. For example, 2𝑥 + 5 = 21
and −7𝑥 − 5 = 9 are both linear equations in one variable. Finding the solution of
the linear equation 𝑎𝑥 + 𝑏 = 𝑐 is simply finding the real value(s) which, when
substituted for 𝑥, makes the left-hand side of the equation equal to the right-hand
side of the equation.

Definition The solution of the equation 𝑎𝑥 + 𝑏 = 𝑐 is the real number(s)

𝒙 which make the equation hold.

Example 1 Verify whether 𝑥 = 5 is a solution of the equation 4𝑥 − 15 = 5

Solution Let’s substitute 𝑥 = 5 in the equation and see whether LHS equals
RHL:

4(𝟓) − 15 = 20 − 15 = 5
Thus 5 is the solution of the linear equation 4𝑥 − 15 = 5.

Example 2 Is 𝑥 = −2 a solution to the linear equation 3𝑥 + 10 = 7?

Solution Let’s substitute 𝑥 = −2 in the equation and see whether LHS equals
RHL:
3(−𝟐) + 10 = −6 + 10 = 4
But, 4 ≠ 7,thus 𝑥 = −2 is not a solution to the equation 3𝑥 + 10 = 7.

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

In order to solve linear equations, we might need to simply the RHS and LHS of
the equation if they involve brackets and parenthesis by performing operation such
as BEDMAS, combing like terms, etc., without out losing the equation-equality
preservation. If the coefficient of the unknown variable is a fraction of the form
𝑎/𝑏 , with 𝑎 ≠ 0 and 𝑏 ≠ 0, we multiply the equation by the reciprocal of 𝑎/𝑏 ,
which is 𝑎/𝑏 , in order to isolate the unknown variable. Whereas, if the linear
equation involves more than one fraction, we can first clear the fractions by
multiplying both sides of the equation by the least common multiple of the
denominators of the fractions 𝐿𝐶𝑀.

Example 3 Solve −19 + 3(𝑥 − 1) + 4𝑥 = −8

Solution First, need to simplify the LHS of the equation:

−19 + 3𝑥 − 3 + 4𝑥 = −8 by distribution of the 3 over the parenthesis

3𝑥 + 4𝑥 = −8 + 19 + 3 by sending −19 and −3 to the RHS
7𝑥 = 14 by combining like terms
𝑥 = 2 by dividing both sides by 7

1 5 11
𝐄𝐱𝐚𝐦𝐩𝐥𝐞 𝟒 Solve for 𝑥 the eqaution (3𝑥 − 1) + (5𝑥 + 1) =
2 3 6
Solution In order to clear the fractions, we need to multiply both sides of the
linear equation by 12 because it is the least common multiple between 2, 3, and 6:
1 5 11
𝟏𝟐 ⋅ (3𝑥 − 1) × 𝟏𝟐 ⋅ (5𝑥 + 1) = 𝟏𝟐 ⋅
2 3 6
(6 × 1)(3𝑥 − 1) + (4 × 5)(5𝑥 + 1) = 11 × 2 by dividing 12 by 2, 3, and 6 resp.

⟹ 6(3𝑥 − 1) + 20(5𝑥 + 1) = 22 ⟺ 18𝑥 − 6 + 100𝑥 + 20 = 22

8 2
⟹ 118𝑥 = 22 + 6 − 20 ⟹ 118𝑥 = 8 ⟹ 𝑥= =
118 59

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

Problem 1 Solve the following linear equations

𝑎) 11𝑧 + 2 = 5(𝑧 − 2) 𝑏) − 63 = 2𝑤 + 9𝑤 − 19

𝑐) − 2(5𝑥 − 6) = 3(−𝑥 − 3) 𝑑) − 3(𝑥 − 4) + 2 = 7 − (𝑥 + 1)

𝑒) − 4[𝑦 − 3(𝑦 − 5)] = 2(6 − 5𝑦) 𝑓) 5(−4𝑥 − 2) = −2(8𝑥 − 1)

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

Problem 2 Solve the following equation (By the method of Clearing fractions)
1 1 7 4 3 1
𝑎) − − 𝑧 = 𝑏) − =− 𝑦−
4 2 2 3 7 2

1 1 1 𝑥−2 𝑥−4 𝑥+4

𝑐) 𝑤 + 𝑤 − 1 = (𝑤 − 4) 𝑑) − = 2+
4 3 2 5 2 10

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

Problem 3 Solve the following equations (By the method of Clearing decimals)
𝑎) 2.2𝑥 + 0.5 = 1.6𝑥 + 0.2 𝑏) 0.04𝑥 + 0.07(10000 − 𝑥) = 625

Problem 4 (Identify contradictions and identities) Solve each of the following

equations and tell whether it is a conditional equation, a contradiction, or an
identity equation.
𝑎) 3[𝑥 − (𝑥 + 1)] = −2 𝑏) 5(3 + 𝑦) + 2 = 2𝑦 + 3𝑦 + 17

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

Section 1.2 Application of linear equations in one variable

Problem 5 (Numbers) The sum of two numbers is 39. One number is three less
than twice the other. What are the two numbers?

Problem 6 (Numbers) The sum of two numbers is 81. One number is one more
than four times the other. What are the two numbers?

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

Problem 7 (Mixture)
A) How many liters of a 40% antifreeze solution must be added to a 4L of a 10%
antifreeze solution to produce a 35% antifreeze solution?

B) Ahmad has 3 L of a 50% antifreeze mixture. How much 75% antifreeze mixture
should he add to get a mixture that is 60% antifreeze?

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

C) How many milliliters of a 2.5% bleach solution must be mixed with a 10%
bleach solution to produce 600 mL of a 5% bleach solution?

Problem 8 (Distance, rate, time) A hiker can hike 1mph faster downhill to Moose
Lake than she can hike uphill back to campsite. If it takes 3 hours to hike to the
lake and 4.5 hours to hike back, what is her speed hiking back to campsite?

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

Section 1.3 Application to geometry and literal equations

Problem 9 (Perimeter Geometry problems) The length of a rectangular corral is 2
ft more than 3 times the width. The corral is situated such that one of its shorter
sides is adjacent to a barn and does not require fencing. If the total amount of
fencing is 774 ft, find the dimensions of the corral.

Problem 10 (Angles) Based on the diagram below, solve for x then find the
measure of each angle.

a)

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

b)

c)

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

d)

e)

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

Problem 11 (Solving a Literal equation Solve the following equation for the
indicated variable.
a) Solve for 𝑦: −2𝑥 + 3𝑦 = 5 b) Solve for 𝑥: 𝑎𝑥 − 3 = 𝑐𝑥 + 7

3 + 𝑝𝑌 1
c) Solve for 𝑌: =𝑏 d) Solve for 𝑏1 : 𝐴 = (𝑏1 + 𝑏2 )ℎ
𝑌 2

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

Problem 12 (Solving applications of Literal equations) Buckingham Fountain is

one of Chicago’s most familiar landmarks. With 133 jets spraying a total of 14,000
gal of water per minute, Buckingham Fountain is one of the world’s largest
fountains. The circumference of the fountain is approximately 880 ft.
a. The circumference of a circle is given by 𝐶 = 2𝜋𝑟. Solve the equation for 𝑟.

b. Use the equation from part (a) to find the radius and diameter of the fountain.
Use 𝜋 = 3.14 and round to the nearest foot.

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

Section 1.4 Solving linear inequalities

A linear inequality states that a linear expression 𝑎𝑥 + 𝑏 is less than, less than or
equal, greater than, or greater than or equal a quantity. Symbolically speaking,
these four situations are denoted as follows:

𝑎𝑥 + 𝑏 < 𝑐 This states that 𝑎𝑥 + 𝑏 is less than 𝑐

𝑎𝑥 + 𝑏 ≤ 𝑐 This states that 𝑎𝑥 + 𝑏 is less than or equal to 𝑐

𝑎𝑥 + 𝑏 > 𝑐 This states that 𝑎𝑥 + 𝑏 is greater than 𝑐

𝑎𝑥 + 𝑏 ≥ 𝑐 This states that 𝑎𝑥 + 𝑏 is greater than or equal to 𝑐

Interval notation and graphical representation of a solution set of an inequality:

Solution set Interval Graph (in red)

notation

𝑥<𝑘 (−∞, 𝑘)

𝑥≤𝑘 (−∞, 𝑘]

𝑥>𝑘 (𝑘, ∞)

𝑥≥𝑘 [ 𝑘, ∞)

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

Example 5 Use interval notation to represent the solution sets then graph.

a) 𝑥 < −1 b) 𝑥 ≥ −4 c) 𝑥 > −2 d) 𝑥 ≤ 3

Solution a) The interval notation of the solution set 𝑥 < −1 is (−∞, −1). The
graph of this solution set is shown right below

b) The interval notation of the solution set 𝑥 ≥ −4 is [4, ∞). The graph of this
solution set is shown right below

c) The interval notation of the solution set 𝑥 > −2 is (−2, ∞). The graph of this
solution set is shown right below

d) The interval notation of the solution set 𝑥 ≤ 3 is (−∞, 3]. The graph of this
solution set is shown in the figure below

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

Example 6 Solve the inequality −7𝑥 + 4(3𝑥 − 1) + 19 ≥ 0 . Express your

solution set in interval notation and graph it.

Solution −7𝑥 + 4(3𝑥 − 1) + 19 ≥ 0

⟹ −7𝑥 + 12𝑥 − 4 + 19 ≥ 0 By removing the bracket of the RHS

⟹ 5𝑥 + 15 ≥ 0 By simplifying the LHS of the inequality

⟹ 5 ≥ −15 By transferring 15 to the RHS of the inequality

⟹ 𝑥 ≥ −5 By dividing both sides of the inequality by 5

In interval notation, the solution set is [−5, ∞). See the figure below for the graph.

Example 7 Solve the inequality −3(2𝑥 + 5) − 9 ≥ 12 . Express your solution

set in interval notation and graph it.

Solution −3(2𝑥 + 5) − 9 < 12

⟹ −6𝑥 − 15 − 9 < 12 By removing the parenthesis in the RHS

⟹ −6𝑥 < 12 + 15 + 9 By transferring −15 and −9 to the LHS

⟹ −6𝑥 < 36 By adding the numbers on the RHS of the inequality

⟹𝑥>6 By dividing both sides by −6 and reversing the inequality sense.

In interval notation, the solution set is (6, ∞). See the figure below for the graph.

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

Problem 13 Write an inequality and an interval notation that represent the

interval shown in the figure below:
a)

b)

c)

d)

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

Problem 14 Solve the linear inequality. Express your answer in interval notation
and graph it.
𝑎) 3𝑥 − 7 > 2(𝑥 − 4) − 1

𝑏) − 2𝑥 − 5 < 2

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

𝑐) − 6(𝑥 − 3) ≥ 2 − 2(𝑥 − 8)

−5𝑥 + 2
𝑑) >𝑥+2
−3

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

𝑒) 4 − 3𝑥 ≥ 10(−𝑥 + 5)

𝑓) 7𝑥 − 35 ≤ −5(4 − 2𝑥)

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

Problem 15 (Application of linear inequalities)

a) Manal received grades of 97, 82, 89, and 99 on her first four algebra tests. To
earn an A in the course, she needs an average of 90 or more. What scores can she
receive on the fifth test to earn an A?

b) The water level in a retention pond in northern California is 72 ft. During a time
of drought, the water level decreases a rate of 0.05 ft/day. The water level 𝐿 (in ft)
is given by the equation 𝐿 = 7.2 − 0.05𝑑 , where 𝑑 is the number of days after
the drought begins. For which days after the beginning of the drought will the
water level be less than 6 ft? Refer to the figure below.

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

Section 1.5 Solving linear inequalities

Problem 16 (Finding the union and intersection of sets)
Given the sets: 𝐴 = {𝑥 |𝑥 < 3} 𝐵 = {𝑥 |𝑥 ≥ −2} 𝐶 = {𝑥|𝑥 ≥ 5}
Graph each of the following sets, then express it in interval notation.

𝑎) 𝐴 ∩ 𝐵 𝑏) 𝐴 ∪ 𝐶

𝑎) 𝐴 ∩ 𝐵

𝑏) 𝐴 ∪ 𝐶

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

Problem 17 (Finding the union and intersection of two intervals)

Find the union or intersection as indicated. Write the answer in interval notation.

𝑎) (−∞, −2) ∪ [−4, 3) 𝑏) (−∞, −2) ∩ [−4, 3)

𝑎) (−∞, −2) ∪ [−4, 3)

𝑏) (−∞, −2) ∩ [−4, 3)

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

Definition An inequality is called a compound inequality if it involves more than

one component or inequality. The use of the logical connectives “or” and “and” is
essential to relate the inequalities. To determine the solution set of a compound
inequality we have to find the set where all the components’ inequalities intersect.

Problem 18 Graph the compound inequality on the number line.

𝑎) 𝑥 > 5 or 𝑥 < −6

𝑏 )𝑥 < −4 or 𝑥 > 8

𝑐) 𝑥 ≥ −4 and 𝑥 < 3

𝑑) 𝑥 < 7 and 𝑥 ≥ 0

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

Problem 19 (Solve a compound inequality: And) Solve the following compound

inequality and graph the solution set. Express your answer in interval notation:

𝑎) − 2𝑥 < 6 and 𝑥+5 ≤7

5 1
𝑏) 𝑥 ≤ 15 and − 𝑥<1
2 2

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

𝑐) − 3𝑥 + 1 ≥ 10 and − 2𝑥 − 5 ≤ 15

𝑑) 4𝑥 + 2 ≥ 6 and 2𝑥 − 2 ≤ −12

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

Problem 20 (Solve a compound inequality: Or) Solve the following compound

inequality and graph the solution set. Express your answer in interval notation:
𝑎) − 3𝑦 − 5 > 4 or 4−𝑦≤6

2 3
𝑏) 𝑥 − 3 ≤ 1 or 𝑥−2>4
3 4

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

1
𝑐) − 5𝑥 − 1 ≥ 24 or 𝑥 − 3 > −1
2

𝑑) 2𝑥 + 5 ≤ 9 or 4𝑥 + 3 ≥ 27

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

Problem 21 (Solve linear inequalities of the form 𝒂 < 𝒙 < 𝒃) Solve the following
inequality and graph the solution set. Express your answer in and interval notation:
𝑎) − 4 < 3𝑥 + 5 ≤ 11

𝑝−2
𝑏) 2 ≥ ≥ −1
−3

6 − 2𝑥
𝑐) − 2 ≤ <2
5

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 ALI SAAD MATH1010-UNIT 1-LINEAR EQUATIONS AND LINEAR INEQUALITIES

References

1- Ali Saad “College Algebra”

2- Sullivan “Algebra & Trigonometry”

3- Julie Miller, Molly O’Neill, and Nancy Hyde “Intermediate

Algebra” McGraw-Hill Create™ (ISBN-13: 9781307766325)

4- MATH1010-ALEKS Instructional Packages for UDST

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