Absolute Value Inequalities Notes
Absolute Value Inequalities - Notes
Name:
Determining the intervals
Case 1: "Less Than"
\( |u| < c \implies -c < u < c \)
All solutions are between these values.
-c 0 +c
Case 2: "Greater Than"
\( |u| > c \implies u < -c \;\text{ OR }\; u > c \)
All solutions are outside of these values.
-c 0 +c
Example 1 \( 2|x + 1| + 5 \ge 9 \)
2|x + 1| + 5 ≥ 9 - 5 - 5
2|x + 1| 2
≥
4 2
|x + 1| ≥ 2
Positive Component
x + 1 ≥ 2 - 1 - 1
x ≥ 1
Negative Component
x + 1 ≤ -2 - 1 - 1
x ≤ -3
\( x \ge 1 \quad\text{OR}\quad x \le -3 \)
-3 0 1
Interval: \( (-\infty, -3] \cup [1, \infty) \)
Example 2 \( -3|x - 2| + 7 > -8 \)
-3|x - 2| + 7 > -8 - 7 - 7
-3|x - 2| -3
>
-15 -3
|x - 2| < 5
-5 < x - 2 < 5 + 2 + 2 + 2
-3 < x < 7
\( -3 < x < 7 \)
-3 2 7
Interval: \( (-3, 7) \)
Special Cases (When constant is negative after isolating)
\( |u| < \text{negative} \) \( \emptyset \) No Solution
\( |u| > \text{negative} \) \( (-\infty, \infty) \) All Real Numbers
Unit 2 - Absolute Value Function
Lesson 3 - Absolute Value Inequalities
Absolute Value Inequalities Practice
Lesson Practice
Absolute Value Inequalities
Name:
For problems 1–3, circle either AND or OR for each inequality:
1. \( |x - 9| < 5 \)
AND
/
OR
2. \( |3x + 1| \ge 13 \)
AND
/
OR
3. \( -2|x + 5| \le -10 \)
AND
/
OR
Solve each inequality, graph the solution set on the number line, and express in interval notation:
4. \( |x + 6| \le 4 \)
Interval:
5. \( |2x - 5| > 9 \)
Interval:
6. \( 3|x - 2| + 4 < 19 \)
Interval:
7. \( -4|2x + 1| - 7 \le -23 \)
Interval:
Unit 2 - Absolute Value Function
Lesson 3 - Absolute Value Inequalities
