🎯 Absolute Value Casework — Breakpoints & Symmetry

Essential technique for solving absolute value equations and inequalities.

🎯 Recognition Cues

• “Solve for $x$” — Equations with absolute value signs
• “Find all real solutions” — Absolute value equations often have multiple cases
• $|x|$, $|x-a|$ — Absolute value expressions
• “For what values” — Absolute value inequalities

📚 Template Solutions

Basic Absolute Value Equation

Multiple Absolute Values

🎯 Worked Examples

Example 1: Basic Absolute Value

Problem: Solve $|x-3| = 7$

Solution:

1. Breakpoint: $x = 3$
2. Case 1 ($x \geq 3$): $x-3 = 7$ → $x = 10$
3. Case 2 ($x < 3$): $-(x-3) = 7$ → $-x+3 = 7$ → $x = -4$
4. Check: $x = 10$: $|10-3| = 7$ ✓; $x = -4$: $|-4-3| = 7$ ✓
5. Answer: $x = 10$ or $x = -4$

Example 2: Multiple Absolute Values

Problem: Solve $|x-1| + |x+2| = 5$

Solution:

1. Breakpoints: $x = 1$ and $x = -2$
2. Create intervals: $x < -2$; $-2 \leq x < 1$; $x \geq 1$
3. Case 1 ($x < -2$): $-(x-1) - (x+2) = 5$ → $-x+1-x-2 = 5$ → $-2x-1 = 5$ → $x = -3$
4. Case 2 ($-2 \leq x < 1$): $-(x-1) + (x+2) = 5$ → $-x+1+x+2 = 5$ → $3 = 5$ (no solution)
5. Case 3 ($x \geq 1$): $(x-1) + (x+2) = 5$ → $x-1+x+2 = 5$ → $2x+1 = 5$ → $x = 2$
6. Check: $x = -3$: $|-3-1| + |-3+2| = 4 + 1 = 5$ ✓; $x = 2$: $|2-1| + |2+2| = 1 + 4 = 5$ ✓
7. Answer: $x = -3$ or $x = 2$

Example 3: Absolute Value Inequality

Problem: Solve $|x-2| < 3$

Solution:

1. Rewrite as compound inequality: $-3 < x-2 < 3$
2. Add 2 to all parts: $-3 + 2 < x < 3 + 2$ → $-1 < x < 5$
3. Answer: $x \in (-1, 5)$

⚠️ Common Pitfalls

Pitfall: Incorrect case analysis

• Fix: Always consider all possible combinations of signs
• Example: $|x-1| + |x+2|$ has 4 possible sign combinations, not 2

Pitfall: Forgetting to check solutions

• Fix: Always substitute solutions back into original equation
• Example: $|x-1| + |x+2| = 5$ might have extraneous solutions

Pitfall: Incorrect inequality direction

• Fix: $|x| < a$ means $-a < x < a$ (and), $|x| > a$ means $x < -a$ or $x > a$ (or)
• Example: $|x| < 2$ is $-2 < x < 2$, not $x < 2$ or $x > -2$

🎯 AMC-Style Worked Example

Problem: Find all real values of $x$ such that $|x^2 - 4| = 3$.

Solution:

1. Set up cases: $x^2 - 4 = 3$ or $x^2 - 4 = -3$
2. Case 1: $x^2 - 4 = 3$ → $x^2 = 7$ → $x = \pm\sqrt{7}$
3. Case 2: $x^2 - 4 = -3$ → $x^2 = 1$ → $x = \pm 1$
4. Check all solutions:
   • $x = \sqrt{7}$: $|(\sqrt{7})^2 - 4| = |7 - 4| = 3$ ✓
   • $x = -\sqrt{7}$: $|(-\sqrt{7})^2 - 4| = |7 - 4| = 3$ ✓
   • $x = 1$: $|1^2 - 4| = |1 - 4| = 3$ ✓
   • $x = -1$: $|(-1)^2 - 4| = |1 - 4| = 3$ ✓
5. Answer: $x = \pm\sqrt{7}$ or $x = \pm 1$

Key insight: Absolute value equations often lead to multiple cases that must all be checked.

🔗 Related Patterns

• Case Analysis — Absolute value problems require careful case analysis
• Breakpoints — Find all values where absolute value arguments equal zero
• Symmetry — Absolute value expressions often have symmetry
• Inequalities — Absolute value inequalities use similar techniques

📝 Practice Checklist

• Master basic absolute value equations
• Practice multiple absolute value problems
• Learn case analysis technique
• Practice absolute value inequalities
• Understand breakpoint identification
• Work on speed and accuracy

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