How to Solve Absolute Value Inequalities

How to Solve Absolute Value Inequalities

A Complete Guide for Students & Teachers

Introduction

"Mastering absolute value inequalities lets you solve real-world problems—like finding safe temperature ranges for vaccines or acceptable error margins in engineering—and ace those tricky exam questions that separate A students from the rest."

What You Need To Know First

Before diving in, ensure you understand: 1. Absolute Value Basics: The absolute value of a number is its distance from zero on the number line, always non-negative. Example: |5| = 5, |-3| = 3. 2. Solving Linear Inequalities: How to solve inequalities like 2x + 3 > 7 and represent solutions on a number line. 3. Compound Inequalities: Combining two inequalities (e.g., -2 < x ≤ 5) and understanding "and" vs. "or" statements.

Key Vocabulary

Formulas To Know

1. Absolute Value Inequality Rules

MEMORISE THIS: - |A| < B (where B > 0) → -B < A < B (compound inequality with "and") - |A| > B (where B > 0) → A < -B or A > B (two separate inequalities with "or") - |A| ≤ B → -B ≤ A ≤ B - |A| ≥ B → A ≤ -B or A ≥ B

What each variable means: - A: Any algebraic expression (e.g., 2x + 3). - B: A positive number (if B is negative, the inequality has no solution or all real numbers as solutions—see Exam Traps).

Step-by-Step Method

Step 1: Isolate the Absolute Value

• Move all terms not inside the absolute value to the other side of the inequality.
• Simplify until the absolute value is alone on one side.

Example: Solve |3x - 2| + 4 ≤ 9. - Subtract 4 from both sides: |3x - 2| ≤ 5.

Step 2: Check the Inequality Symbol

• If the symbol is < or ≤, rewrite as a compound inequality (-B < A < B or -B ≤ A ≤ B).
• If the symbol is > or ≥, rewrite as two separate inequalities (A < -B or A > B).

Example (continued): |3x - 2| ≤ 5 → -5 ≤ 3x - 2 ≤ 5.

Step 3: Solve the Compound or Separate Inequalities

• For compound inequalities, solve all parts simultaneously.
• For separate inequalities, solve each one individually.

Example (continued): - Add 2 to all parts: -5 + 2 ≤ 3x ≤ 5 + 2 → -3 ≤ 3x ≤ 7. - Divide by 3: -1 ≤ x ≤ 7/3.

Step 4: Graph the Solution (If Required)

• For ≤ or ≥, use closed circles on the number line.
• For < or >, use open circles.
• Shade the region between the points for "and" solutions; shade both directions for "or" solutions.

Example (continued): Graph -1 ≤ x ≤ 7/3 with closed circles at -1 and 7/3, shading in between.

Step 5: Write the Solution in Interval Notation

• Use parentheses ( ) for open circles (strict inequalities).
• Use brackets [ ] for closed circles (inclusive inequalities).

Example (continued): Solution is [-1, 7/3].

Worked Examples

Example 1 - Basic

Problem: Solve |x - 4| < 3.

Step 1: Absolute value is already isolated. Step 2: Rewrite as compound inequality: -3 < x - 4 < 3. Step 3: Add 4 to all parts: 1 < x < 7. Step 4: Graph with open circles at 1 and 7, shading in between. Step 5: Interval notation: (1, 7).

What we did and why: We used the rule |A| < B → -B < A < B to split the inequality into two parts. Solving both simultaneously gives the range of x-values that satisfy the original inequality.

Example 2 - Medium

Problem: Solve |2x + 1| ≥ 5.

Step 1: Absolute value is already isolated. Step 2: Rewrite as two separate inequalities: 2x + 1 ≤ -5 or 2x + 1 ≥ 5. Step 3:
- For 2x + 1 ≤ -5: Subtract 1 → 2x ≤ -6 → x ≤ -3.
- For 2x + 1 ≥ 5: Subtract 1 → 2x ≥ 4 → x ≥ 2. Step 4: Graph with closed circles at -3 and 2, shading left of -3 and right of 2. Step 5: Interval notation: (-∞, -3] ∪ [2, ∞).

What we did and why: We used the rule |A| ≥ B → A ≤ -B or A ≥ B to split the inequality into two cases. Solving each case separately gives the solution set where x is either less than or equal to -3 or greater than or equal to 2.

Example 3 - Exam Style

Problem: Solve 3|x - 2| - 1 > 8. Write the solution in interval notation.

Step 1: Isolate the absolute value: Add 1 to both sides → 3|x - 2| > 9. Step 2: Divide by 3 → |x - 2| > 3. Step 3: Rewrite as two inequalities: x - 2 < -3 or x - 2 > 3. Step 4:
- For x - 2 < -3: Add 2 → x < -1.
- For x - 2 > 3: Add 2 → x > 5. Step 5: Interval notation: (-∞, -1) ∪ (5, ∞).

What we did and why: We first isolated the absolute value by undoing the operations outside it. Then, we applied the rule for |A| > B to split the inequality into two cases. The solution is all x-values less than -1 or greater than 5.

Common Mistakes

Exam Traps

1-Minute Recap

"Alright, let’s lock this in—tonight or the morning of your exam. Absolute value inequalities come in two flavors: less-than and greater-than. Here’s the cheat sheet:

1. Isolate the absolute value first. Get it alone on one side.
2. If it’s < or ≤, rewrite it as a compound inequality: -B < A < B. Solve it all at once.
3. If it’s > or ≥, split it into two separate inequalities: A < -B or A > B. Solve each one.
4. Check B. If B is negative or zero, the rules change—no solution for <, all real numbers for ≥.
5. Graph it. Open circles for < or >, closed for ≤ or ≥. Shade between for "and," both ways for "or."

That’s it. Practice two problems of each type, and you’ll own this topic. You’ve got this!