Solve absolute value equations and inequalities

1. Absolute Value Equations
Chapter 1 Equations and Inequalities

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

3. Absolute Value
 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 (cont.)
 The solution set for the equation must include
both a and –a.
 Example: Solve
x a
 9 4 7x

5. Absolute Value
 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
 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
 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
 College Algebra
 Page 163: 10-22 (evens), page 155: 26-36 (evens),
page 144: 36-44 (4), 78