This document discusses how to solve absolute value equations and inequalities. It explains that absolute value equations have solution sets that include both the positive and negative values that satisfy the equation. Absolute value inequalities can be solved using the properties that absolute values are less than a number if the value is between the negatives of the number, and greater than a number if it is less than the negative or greater than the positive of the number. The document provides examples of solving absolute value equations and inequalities and discusses special cases when the absolute value expression is always true, false, or can be treated as a normal equation.

1. Absolute Value Equations &
Inequalities
Chapter 1 Equations and Inequalities

2. Concepts and Objectives
⚫ Solve absolute value equations and inequalities

3. Absolute Value Equations
⚫ You should recall that the absolute value of a number a,
written |a|, gives the distance from a to 0 on a number
line.
⚫ By this definition, the equation |x| = 3 can be solved by
finding all real numbers at a distance of 3 units from 0.
Both of the numbers 3 and ‒3 satisfy this equation, so
the solution set is {‒3, 3}.

4. Absolute Value Equations (cont.)
⚫ The solution set for the equation must include
both a and –a.
⚫ Example: Solve
=x a
− =9 4 7x

5. Absolute Value Equations (cont.)
⚫ The solution set for the equation must include
both a and –a.
⚫ Example: Solve
The solution set is
=x a
− =9 4 7x
− =9 4 7x − = −9 4 7x
− = −4 2x − = −4 16x
=
1
2
x = 4x
or
 
 
 
1
,4
2

6. Absolute Value Inequalities
⚫ For absolute value inequalities, we make use of the
following two properties:
⚫ |a| < b if and only if –b < a < b.
⚫ |a| > b if and only if a < –b or a > b.
⚫ Example: Solve − + 5 8 6 14x

7. Absolute Value Inequalities (cont.)
⚫ Example: Solve
The solution set is
− + 5 8 6 14x
or
− 5 8 8x
−  −5 8 8x − 5 8 8x
−  −8 13x − 8 3x

13
8
x −
3
8
x
   
− −    
   
3 13
, ,
8 8

8. Special Cases
⚫ Since an absolute value expression is always
nonnegative:
⚫ Expressions such as |2 – 5x| > –4 are always true. Its
solution set includes all real numbers, that is, (–, ).
⚫ Expressions such as |4x – 7| < –3 are always false—
that is, it has no solution.
⚫ The absolute value of 0 is equal to 0, so you can solve
it as a regular equation.

9. Classwork
⚫ 1.8 Assignment (College Algebra)
⚫ Page 163: 10-22 (even), page 155: 36-52 (4),
page 144: 72-88 (4)
⚫ 1.8 Classwork Check
⚫ Quiz 1.7
Your six weeks test is next Friday.