🎯 Absolute Value — Piecewise Setup & Distance View

Essential technique for handling absolute value expressions in AMC problems.

🎯 Key Ideas

Absolute Value Definition — $|x| = \begin{cases} x & \text{if } x \geq 0 \ -x & \text{if } x < 0 \end{cases}$

Distance Interpretation — $|x - a|$ represents the distance between $x$ and $a$ on the number line.

Piecewise Approach — Break absolute value expressions into cases based on the sign of the argument.

📊 Essential Techniques

Basic Properties

Solving Strategies

🎯 Micro-Examples

Example 1: Solve $|x - 3| = 7$

• Method 1: $x - 3 = 7$ or $x - 3 = -7$
• Solve: $x = 10$ or $x = -4$
• Answer: $x = 10$ or $x = -4$

Example 2: Solve $|2x + 1| < 5$

• Rewrite: $-5 < 2x + 1 < 5$
• Subtract 1: $-6 < 2x < 4$
• Divide by 2: $-3 < x < 2$
• Answer: $x \in (-3, 2)$

Example 3: Solve $|x - 2| + |x + 1| = 5$

• Breakpoints: $x = 2$ and $x = -1$
• Case 1 ($x < -1$): $-(x-2) - (x+1) = 5$ → $-2x + 1 = 5$ → $x = -2$
• Case 2 ($-1 \leq x < 2$): $-(x-2) + (x+1) = 5$ → $3 = 5$ (no solution)
• Case 3 ($x \geq 2$): $(x-2) + (x+1) = 5$ → $2x - 1 = 5$ → $x = 3$
• Answer: $x = -2$ or $x = 3$

⚠️ Common Traps & Fixes

Trap: Forgetting to check both cases

• Fix: Always consider both positive and negative cases
• Example: $|x| = 3$ has solutions $x = 3$ and $x = -3$

Trap: 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 $x > -2$

Trap: Missing breakpoints in complex expressions

• Fix: Find all values where any absolute value argument equals zero
• Example: $|x-1| + |x+2|$ has breakpoints at $x = 1$ and $x = -2$

🎯 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. Verify:
   • $|(\sqrt{7})^2 - 4| = |7 - 4| = 3$ ✓
   • $|(-\sqrt{7})^2 - 4| = |7 - 4| = 3$ ✓
   • $|1^2 - 4| = |1 - 4| = 3$ ✓
   • $|(-1)^2 - 4| = |1 - 4| = 3$ ✓
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 Topics

• Linear Inequalities — Absolute value inequalities are special cases
• Quadratic Equations — Absolute value can involve quadratics
• Piecewise Functions — Absolute value is a piecewise function
• Distance — Absolute value represents distance on number line

📝 Practice Checklist

• Master basic absolute value equations
• Practice absolute value inequalities
• Learn piecewise approach
• Practice with multiple absolute values
• Understand distance interpretation
• Work on speed and accuracy

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