1. A -3 ≥ x ≥ 1 B -3 > x > 1 C -3 < x < 1 D -3 ≤ x ≤ 130
Sec
Which compound inequality is shown by the graph? 1 –3 –2 –1 3 2 –4 4 5 6 –5 –6
A -3 ≥ x ≥ 1
B -3 > x > 1
C -3 < x < 1
D -3 ≤ x ≤ 1

2. A x ≥ 0 or x ≤ -5 B x > 0 or x ≤ -5 C x ≤ 0 or x ≥ -530
Sec
Which compound inequality is shown by the graph? 1 –3 –2 –1 3 2 –4 4 5 6 –5 –6
A x ≥ 0 or x ≤ -5
B x > 0 or x ≤ -5
C x ≤ 0 or x ≥ -5
D x < 0 or x ≤ -5

3. Solving absolute Value inequality
Lesson 2-7
Solving absolute Value inequality
You'll Learn how to
Solve absolute value inequality

4. Absolute value
The absolute value of a number is the distance of that number from zero
EXAMPLE 1
| x | = 7
Read as
“The distance of my house from the city center is 7”
= 7 or
= -7 1 2 3 4 5 6 7 -1 -2 -3 -4 -5 -6 -7
center
EXAMPLE 2
Solve and graph
|x + 4| = 2
x + 4 = 2 or
x + 4 = -2
x = -2
x = -6 1 2 3 4 5 6 7 -1 -2 -3 -4 -5 -6 -7

5. 3 4 Case1 “and” | x | < 5 < 5 and > -5 Solve the inequality
EXAMPLE 3
| x | < 5
Read as
“The distance of my house from the city center is < 5”
< 5
and
> -5 1 2 3 4 5 6 7 -1 -2 -3 -4 -5 -6 -7
center
EXAMPLE 4
Solve the inequality
|x| – 3 < – 2
ISOLATE the absolute value expression
|x| < 1
Write as compound inequality
x < 1
and
x > -1 1 2 3 4 5 6 7 -1 -2 -3 -4 -5 -6 -7

6. 5 6 Case2 “OR” | x | > 3 > 3 or < -3 Solve the inequality
EXAMPLE 5
| x | > 3
Read as
“The distance of my house from the city center is > 3”
> 3 or
< -3 1 2 3 4 5 6 7 -1 -2 -3 -4 -5 -6 -7
EXAMPLE 6
Solve the inequality
|x| + 14 ≥ 19
ISOLATE the absolute value expression
|x| ≥ 5
Write as compound inequality
x ≥ 5 or
x ≤ -5 1 2 3 4 5 6 7 -1 -2 -3 -4 -5 -6 -7

7. 7 Special Cases: Solve and graph a) |x| + 5 < 2 |x| < -3
EXAMPLE 7
Solve and graph
a) |x| + 5 < 2
ISOLATE the absolute value expression
|x| < -3
Think:
distance is never < zero
Conclusion:
No Solution
b) |x| + 3 > 2
ISOLATE the absolute value expression
|x| > -1
Think:
distance is always > zero
Conclusion:
Solution is all real numbers
Lesson Summary
|x| < a (positive)
x < a and x > -a
|x| < a (negative)
No Solution
|x| > a (positive)
x > a or x < -a
|x| > a (negative)
Solution is all real numbers

8. Solve and graph a) |x| – 3 < –1 b) |x – 1| ≤ 2 |x| < 2 x – 1 ≤ 2
Exercises
Solve and graph
a) |x| – 3 < –1
b) |x – 1| ≤ 2
ISOLATE
|x| < 2
Split
x – 1 ≤ 2
and
x – 1 ≥ -2
x < 2
and
x > -2
c) |x| + 12 > 13
d) 3 + |x + 2| > 5
ISOLATE
|x| > 1
|x + 2| > 2
Split
x > 1 or
x < -1
x + 2 > 2 or
x + 2 < -2
e) |x + 4| – 5 > –8
f) |x + 2| + 9 < 7
ISOLATE
|x + 4| > -3
|x + 2| < -2
Solution is all real #s
No Solution