Solving Absolute Value Inequalities

Absolute value inequalities involve finding the range of values that satisfy an inequality involving absolute value. This concept is important for understanding more complex algebraic and real-world problems.

Outline

Introduction to Absolute Value Inequalities
Solving Absolute Value Inequalities with Less Than
Solving Absolute Value Inequalities with Greater Than
Examples and Practice Exercises
Homework
Revision
YouTube Video Explanation

Examples

Example 1

Solve the inequality \( | x | < 4 \).

The solution is \( -4 < x < 4 \).

Example 2

Solve the inequality \( | x - 3 | \leq 5 \).

The solution is \( -2 \leq x \leq 8 \).

Example 3

Solve the inequality \( | 2x + 1 | > 7 \).

The solution is \( x < -4 \) or \( x > 3 \).

Example 4

Solve the inequality \( | 3x - 2 | \geq 4 \).

The solution is \( x \leq -\frac{2}{3} \) or \( x \geq 2 \).

Example 5

Solve the inequality \( | x/2 - 1 | < 3 \).

The solution is \( -4 < x < 8 \).

Example 6

Solve the inequality \( | 5 - x | \geq 2 \).

The solution is \( x \leq 3 \) or \( x \geq 7 \).

Exercises

Exercise 1

Solve the inequality \( | x + 4 | < 6 \).

Solution: \( -10 < x < 2 \).

Exercise 2

Solve the inequality \( | 2x - 3 | \leq 5 \).

Solution: \( -1 \leq x \leq 4 \).

Exercise 3

Solve the inequality \( | 4 - x | > 3 \).

Solution: \( x < 1 \) or \( x > 7 \).

Exercise 4

Solve the inequality \( | 3x + 2 | \geq 6 \).

Solution: \( x \leq -\frac{8}{3} \) or \( x \geq \frac{4}{3} \).

Exercise 5

Solve the inequality \( | x/3 + 5 | < 2 \).

Solution: \( -11 < x < -1 \).

Exercise 6

Solve the inequality \( | 6 - x | \geq 4 \).

Solution: \( x \leq 2 \) or \( x \geq 10 \).

Homework

Complete the following problems for additional practice:

Solve \( | x - 5 | < 7 \)
Solve \( | 2x + 3 | \geq 8 \)
Solve \( | 4 - x | \leq 5 \)
Solve \( | x/2 + 4 | > 3 \)
Solve \( | 3x - 7 | < 2 \)

Revision

Review these important points:

When solving \( | A | < B \), the solution is \( -B < A < B \).
When solving \( | A | \leq B \), the solution is \( -B \leq A \leq B \).
When solving \( | A | > B \), the solution is \( A < -B \) or \( A > B \).
When solving \( | A | \geq B \), the solution is \( A \leq -B \) or \( A \geq B \).

Video Explanation

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