Algebra 101 - Unit 2

Lesson 12: Absolute Value Equations and Inequalities

The absolute value of a number is its distance from 0 on the number line, regardless of direction. Example: \(|-5| = 5\) and \(|5| = 5\).

That means absolute value is always non-negative.

Absolute Value Equations

To solve equations with absolute value, we split into two cases.

1. \(|A| = B\) becomes \(A = B\) or \(A = -B\), as long as \(B \ge 0\).
2. If \(B < 0\), there is no solution.

Example 1:

Solve: \(|x| = 7\)

Split into two cases: \(x = 7\) or \(x = -7\).

Solution: \(\{ -7, 7 \}\)

Example 2:

Solve: \(|x - 3| = 5\)

• Case 1: \(x - 3 = 5 \Rightarrow x = 8\)
• Case 2: \(x - 3 = -5 \Rightarrow x = -2\)

Solution: \(\{ -2, 8 \}\)

Absolute Value Inequalities

Solving inequalities depends on whether it’s a “less than” or “greater than” situation.

• Less than (|A| < B): Means \(-B < A < B\) → an AND inequality.
• Greater than (|A| > B): Means \(A < -B\) or \(A > B\) → an OR inequality.

Example 3 (Less Than):

Solve: \(|x| < 4\)

Rewrite as: \(-4 < x < 4\)

Solution: all numbers between -4 and 4 (not including).

Example 4 (Greater Than):

Solve: \(|x - 2| \ge 3\)

• Case 1: \(x - 2 \ge 3 \Rightarrow x \ge 5\)
• Case 2: \(x - 2 \le -3 \Rightarrow x \le -1\)

Solution: \(x \le -1\) or \(x \ge 5\)

Practice Problems

1. Solve: \(|x + 1| = 6\)
2. Solve: \(|2x - 3| = 5\)
3. Solve: \(|x| \le 2\)
4. Solve: \(|x - 4| > 7\)

Lesson Summary

• Absolute value measures distance from 0 (always non-negative).
• \(|A| = B\) → two cases: \(A = B\) or \(A = -B\) (if \(B \ge 0\)).
• \(|A| < B\) → double inequality (AND).
• \(|A| > B\) → split inequality (OR).