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1.8: Solving Absolute Value Equations and Inequalities: Section Outline

This document discusses solving absolute value equations and inequalities. It begins by explaining how to solve absolute value equations using properties of absolute values. Examples are provided. It then explains how to solve absolute value inequalities using similar properties. Special cases for inequalities where the constant is not positive are also discussed. The document gives examples of solving absolute value equations and inequalities as well as using them to describe distances on a number line. Exercises with solutions are provided at the end.

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AHMED ALSHAMMARI
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16K views3 pages

1.8: Solving Absolute Value Equations and Inequalities: Section Outline

This document discusses solving absolute value equations and inequalities. It begins by explaining how to solve absolute value equations using properties of absolute values. Examples are provided. It then explains how to solve absolute value inequalities using similar properties. Special cases for inequalities where the constant is not positive are also discussed. The document gives examples of solving absolute value equations and inequalities as well as using them to describe distances on a number line. Exercises with solutions are provided at the end.

Uploaded by

AHMED ALSHAMMARI
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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1.8: SOLVING ABSOLUTE VALUE EQUATIONS AND INEQUALITIES ABUJIYA, M.R.

1.8: SOLVING ABSOLUTE VALUE EQUATIONS


AND INEQUALITIES

Section Outline:
▪ Absolute value equations ▪ Absolute value inequalities ▪ Special cases inequalities
▪ Absolute value model for distance

I. Absolute Value Equations

The absolute value of a number a is defined in section P.2 as a   a . To solve absolute value
equations, we use the following properties:

Prosperities of Absolute Value Equations


If b  0 , then a b if and only if a  b or a  b
If b R , then a b if and only if a  b or a  b

Example 1: Solve each equation.


3
(a) 9  4x  7 (b) 2 2  6 (c) 2x  3  4  5x (d) x2  x
2x 1

(e) 3 x 2  x  18 5 1  x  6 x 1  8
2
(f)

II. Absolute Value Inequalities

Absolute value inequalities can be solved using the following properties:

Prosperities of Absolute Value Inequalities


If b  0 , then a b if and only if b  a  b
a b if and only if a  b or a  b

Example 2: Solve each inequality.

1 1 1 2 1
(a) 4x  6  10 (b) 5 x  3  15 (c) 1 4 (d) 4x   
2x  3 2 3 3 6

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1.8: SOLVING ABSOLUTE VALUE EQUATIONS AND INEQUALITIES ABUJIYA, M.R.

1
(e) 2  x  1  3 (f) 0  x  5 
2

III. Special Cases Inequalities

In the three of the above properties, the constant b is required to be positive. However, if b  0 ,
use the fact that a  0 and consider the truth of the statement. The following properties can serve
as a guide.

More Prosperities of Absolute Value Equations and Inequalities


If b  0 , a  b is undefined ; a  0 a  0
a  b is undefined ; a  0  no solution
a  b is undefined ; a  0 a  0
a  b is always TRUE ; a  0  a   ,  
a  b is always TRUE ; a  0  a   , 0    0,  

Example 3: Solve each equation or inequality.

(a) 3x  7  1   2 (b) 2 8  4 x   2 (c) 28  7 x  0 (d) 2x  1  0

x4
(e) 3 3x  2  2  2 (f) 0
3x  1

IV. Absolute Value Model for Distance

Absolute value equations or inequalities can be used to express distances. Recall in section P.2, we
define the distance between two numbers a and b, on a real number line to be
a b or b a

Example 4: Use the absolute value to describe the given statement.

(a) all real number x less than 3 units from 2.


(b) all real number x more than 2 units from 5.
(c) all real number x at least 5 units from 7.
(d) all real number x at most 4 units from 2.

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1.8: SOLVING ABSOLUTE VALUE EQUATIONS AND INEQUALITIES ABUJIYA, M.R.

___________ Exercises ___________

1. Determine the solution set of each equation.

(a)  x  x (b) x  x (c)  x  9

2. If 2 x  7  3 is equivalent to a  3x  2  b . Find the value of a  b

3. Solve the absolute value inequality x 4  2x 2  1  0

4. Solve the absolute value inequality x  1  3 x  2

5. Find the solution set of absolute value equation 4x 2  23x  6  0

Find the number of solutions of the equation 2 x  1  5 2 x  1  4 1  2 x  0


3 2
6.

1 5
7. Solve the absolute value equation 8  5 x  33 .
3 6

3
8. Solve the absolute value inequality  5  3.
2x  7

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