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
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,
u  a is equivalent to u  a or u  a
Solve: 2 x  3  11
2 x  3  11
2 x  3  11 or 2 x  3  11
2 x  14
2 x  8
x7
x  4
Solution set: {-4, 7}
Theorem
u  a is equivalent to
aua
u  a is equivalent to
aua
In other words, |u| < a is equivalent
to -a < u and u < a.
Ex.
1. Draw the “river”
2. Subtract 21 from
both sides
3. Simplify
4. Check your answer
5. Graph the solution
-15
Solve y + 21 ≥ 7
- 21 -21
y ≥ -14
(-14) + 21 ≥ 7
7≥7
●
-14
-13
Ex.
1.
2.
3.
4.
5.
6.
7.
8.
Solve
Draw “the river”
Subtract 8y from both
sides
Simplify
Add 14 to both sides
Simplify
Rewrite inequality
with the variable first
Check your answer
Graph the solution
16
8y + 3 > 9y - 14
- 8y
- 8y
3 > y - 14
+ 14 + 14
17 > y
y < 17
8(16) + 3 > 9(16) – 14
131 > 130
o
17
18
Ex.
Solve
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.
16
8y + 3 > 9y - 14
- 8y
- 8y
3 > y - 14
+ 14 + 14
17 > y
y < 17
8(16) + 3 = 9(16) – 14
131 > 130
o
17
18
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.
Absolute Value Inequality
Solve | x – 3 | < 5
(1) The absolute value expression is isolated
(2) It is a well defined absolute value inequality
(3) It is a “less thand” inequality. (go to the jail)
a) set jail boundaries: –5, and 5
b) write compound inequalities: (stay in between the jail
boundaries)
–5 < x – 3 < 5
+3
+3
–2 < x < 8
+3
Absolute Value Inequality
You try this!
Solve | x + 4 | < 1
(1) The absolute value expression is isolated
(2) It is a well defined absolute value inequality
(3) It is a “less thand” inequality. (go to the jail)
a) set jail boundaries: –1, and 1
b) write compound inequalities: (stay in between the jail
boundaries)
–1 < x + 4 < 1
–4
–4
–4
–5 < x < –3
Absolute Value Inequality
Solve | 2x + 3 | – 3 ≥ 2
(1) The absolute value expression is NOT isolated
| 2x + 3 | – 3 ≥ 2
+3 +3
| 2x + 3 | ≥ 5
(2) It is a well defined absolute value inequality
(3) It is a “greator” inequality. (leaving the jail)
a) set jail boundaries: –5, and 5
b) write compound inequalities: (stay left to left boundary
and right to the right boundary)
2x + 3 ≤ –5
–3 –3
2x ≤ –8
or
x ≤ –4
2x + 3 ≥ 5
–3 –3
or
2x ≥ 2
or
x≥1
Absolute Value Inequality
You try this!
Solve | 4x – 3 | + 5 ≥ 8
(1) The absolute value expression is NOT isolated
| 4x – 3| + 5 ≥ 8
–5 –5
| 4x – 3 ≥ 3
(2) It is a well defined absolute value inequality
(3) It is a “greator” inequality. (leaving the jail)
a) set jail boundaries: –3, and 3
b) write compound inequalities: (stay left to left boundary
and right to the right boundary)
4x – 3 ≤ –3
+3 +3
4x ≤ 0
x≤0
or
or
or
4x – 3 ≥ 3
+3 +3
4x ≥ 6
x ≥ 3/2
Summary
1. Before solving absolute value equations or inequalities, you
MUST isolate the absolute value expression and check them
are defined or not.
2. Remember telling yourself the jail story. It will help you set
up the correct equations or compound inequalities.
3. 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.
Solve: 3x  1  5
3x  1  5
 5  3x  1  5
 4  3x  6
4
 x2
3
4


Solution Set:  x |   x  2
3


or
 4 
  ,2 
 3 
Theorem
If a is any positive number, then
u  a is equivalent to u   a or u  a
u  a is equivalent to u   a or u  a
Solve: 4 x  3  15
4 x  3  15
4 x  3  15 or
4 x  3  15
4 x  18
4 x  12
9
x
2
x3
9


x | x   or x  3
2


9

 ,  or [3, )
2
