1. College Algebra: Section 1
College Algebra: Section 1.6 Equations and Inequalities Involving Absolute Value
Objectives of this Section
Solve Equations Involving Absolute Value
Solve Inequalities Involving Absolute Value

2. Equations Involving Absolute Value
If the absolute value of an expression equals some positive number a, then the expression itself equals either a or -a. Thus,

4. Theorem

5. Ex. Solve y + 21 ≥ 7 y ≥ -14 (-14) + 21 ≥ 7 7 ≥ 7 ● - 21 -21 -15 -14
y ≥ -14
(-14) + 21 ≥ 7
7 ≥ 7
Draw the “river”
Subtract 21 from both sides
Simplify
Check your answer
Graph the solution ●
-15
-14
-13

6. Ex. Solve 8y + 3 > 9y - 14 - 8y - 8y 3 > y - 14 + 14 + 14
17 > y
y < 17
8(16) + 3 > 9(16) – 14
131 > 130
Draw “the river”
Subtract 8y from both sides
Simplify
Add 14 to both sides
Rewrite inequality with the variable first
Check your answer
Graph the solution o 16 17 18

7. Ex. Solve 8y + 3 > 9y - 14 - 8y - 8y 3 > y - 14 + 14 + 14
17 > y
y < 17
8(16) + 3 = 9(16) – 14
131 > 130
Big Tip!!!
At the end of solving inequality, always put the variable at the LEFT hand side. Then arrow of the inequality sign tells you the correct graph
y < 17
The graph should toward to the left. o 16 17 18

8. To Solve the Absolute Value Inequalities
1. Isolate the absolute value expression.
2. Make sure the absolute value inequality can be defined.
3. For any defined absolute value inequality with template:
|X|  a, where a > 0 and  = <, >, ≤, ≥
write the corresponding compound inequalities by following the
rules:
a) if “|X| > a” or “|X| ≥ a”, meaning “greator”, “leaving the jail”
b) set “jail boundaries” as “–a” and “a”
c) write compound inequalities as
X < –a or X > a meaning “stay left to the left
boundary or right to the right boundary”
d) if “|X| < a” or “|X| ≤ a”, meaning “less thand”, “going
to the jail”
e) set “jail boundaries” as “–a” and “a”
f) write compound inequalities as
–a < X < a (and) meaning “stay between the two
boundaries”
4. Solve the converted compound inequalities.

9. –2 < x < 8 Absolute Value Inequality Solve | x – 3 | < 5
The absolute value expression is isolated
It is a well defined absolute value inequality
It is a “less thand” inequality. (go to the jail)
set jail boundaries: –5, and 5
write compound inequalities: (stay in between the jail boundaries)
–5 < x – 3 < 5
–2 < x < 8

10. –5 < x < –3 Absolute Value Inequality You try this!
Solve | x + 4 | < 1
The absolute value expression is isolated
It is a well defined absolute value inequality
It is a “less thand” inequality. (go to the jail)
set jail boundaries: –1, and 1
write compound inequalities: (stay in between the jail boundaries)
–1 < x + 4 < 1
–5 < x < –3
– – 4 – 4

11. Absolute Value Inequality
Solve | 2x + 3 | – 3 ≥ 2
The absolute value expression is NOT isolated
| 2x + 3 | – 3 ≥ 2
| 2x + 3 | ≥ 5
It is a well defined absolute value inequality
It is a “greator” inequality. (leaving the jail)
set jail boundaries: –5, and 5
write compound inequalities: (stay left to left boundary and right to the right boundary)
2x + 3 ≤ – or 2x + 3 ≥ 5
2x ≤ – or 2x ≥ 2
x ≤ – or x ≥ 1
– – 3
– – 3

12. Absolute Value Inequality
You try this!
Solve | 4x – 3 | + 5 ≥ 8
The absolute value expression is NOT isolated
| 4x – 3| + 5 ≥ 8
| 4x – 3 ≥ 3
It is a well defined absolute value inequality
It is a “greator” inequality. (leaving the jail)
set jail boundaries: –3, and 3
write compound inequalities: (stay left to left boundary and right to the right boundary)
4x – 3 ≤ – or 4x – 3 ≥ 3
4x ≤ or 4x ≥ 6
x ≤ or x ≥ 3/2
– – 5

13. Summary
Before solving absolute value equations or inequalities, you MUST isolate the absolute value expression and check them are defined or not.
Remember telling yourself the jail story. It will help you set up the correct equations or compound inequalities.
Solve the equations or compound inequalities and don’t forget the learned knowledge such as when multiplying or dividing a negative number, you need flip the inequality sign.

14. Solution Set:

15. Theorem