📊 Equations & Inequalities

Mastering equation solving and inequality techniques is crucial for AMC success. This includes absolute value, systems, and advanced inequality methods.

🎯 Key Ideas

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

Systems: Multiple equations with multiple variables. Common methods: substitution, elimination, graphing.

Inequalities: AM-GM inequality and other techniques for optimization problems.

📐 Absolute Value Equations

Basic Strategy

  1. Isolate the absolute value expression
  2. Split into two cases: positive and negative
  3. Solve each case separately
  4. Check solutions in original equation

Example: $|2x - 3| = 7$

  1. Already isolated: $|2x - 3| = 7$
  2. Split: $2x - 3 = 7$ or $2x - 3 = -7$
  3. Solve: $2x = 10$ or $2x = -4$
  4. Solutions: $x = 5$ or $x = -2$
  5. Check: $|2(5) - 3| = |7| = 7$ ✓ and $|2(-2) - 3| = |-7| = 7$ ✓

Pitfall: Extraneous Solutions

Always check solutions! Some may not work in the original equation.

🔄 Systems of Equations

Substitution Method

  1. Solve one equation for one variable
  2. Substitute into other equation(s)
  3. Solve for remaining variable(s)
  4. Back-substitute to find all variables

Elimination Method

  1. Multiply equations to get same coefficients
  2. Add/subtract to eliminate one variable
  3. Solve for remaining variable(s)
  4. Back-substitute

Example: Solve $\begin{cases} x + y = 5 \ 2x - y = 1 \end{cases}$

By Elimination:

  1. Add equations: $(x + y) + (2x - y) = 5 + 1$
  2. Simplify: $3x = 6$, so $x = 2$
  3. Substitute: $2 + y = 5$, so $y = 3$
  4. Solution: $(2, 3)$

⚖️ AM-GM Inequality

Statement: For positive real numbers $a_1, a_2, \ldots, a_n$: $$\frac{a_1 + a_2 + \cdots + a_n}{n} \geq \sqrt[n]{a_1 a_2 \cdots a_n}$$

Equality: Holds when all numbers are equal.

Applications

  • Optimization: Find maximum/minimum values
  • Proofs: Establish bounds and relationships
  • Problem Solving: Recognize when AM-GM applies

Example: Find minimum of $x + \frac{1}{x}$ for $x > 0$

Solution:

  1. Apply AM-GM to $x$ and $\frac{1}{x}$: $$\frac{x + \frac{1}{x}}{2} \geq \sqrt{x \cdot \frac{1}{x}} = \sqrt{1} = 1$$
  2. Multiply by 2: $x + \frac{1}{x} \geq 2$
  3. Equality when $x = \frac{1}{x}$, so $x = 1$
  4. Minimum value is 2, achieved when $x = 1$

🎯 AMC-Style Worked Example

Problem: Find all real solutions to $|x^2 - 4| = 3x$.

Solution:

  1. Case 1: $x^2 - 4 \geq 0$ (i.e., $x \leq -2$ or $x \geq 2$)

    • Then $|x^2 - 4| = x^2 - 4$
    • Equation: $x^2 - 4 = 3x$
    • Rearrange: $x^2 - 3x - 4 = 0$
    • Factor: $(x-4)(x+1) = 0$
    • Solutions: $x = 4$ or $x = -1$
    • Check constraints: $x = 4$ works (since $4 \geq 2$), but $x = -1$ doesn’t (since $-1$ is not $\leq -2$ or $\geq 2$)

  2. Case 2: $x^2 - 4 < 0$ (i.e., $-2 < x < 2$)

    • Then $|x^2 - 4| = -(x^2 - 4) = 4 - x^2$
    • Equation: $4 - x^2 = 3x$
    • Rearrange: $x^2 + 3x - 4 = 0$
    • Factor: $(x+4)(x-1) = 0$
    • Solutions: $x = -4$ or $x = 1$
    • Check constraints: $x = 1$ works (since $-2 < 1 < 2$), but $x = -4$ doesn’t (since $-4$ is not in $(-2, 2)$)

  3. Final Answer: $x = 4$ and $x = 1$

🔍 Common Traps & Fixes

Trap: Forgetting to check case constraints

Fix: Always verify that solutions satisfy the original case conditions.

Trap: Extraneous solutions from squaring

Fix: Check all solutions in the original equation.

Trap: Wrong AM-GM application

Fix: Ensure all numbers are positive before applying AM-GM.

Trap: Missing solutions in systems

Fix: Check all possible combinations and don’t assume unique solutions.

📋 Quick Reference

Absolute Value Properties

  • $|a| \geq 0$ (always non-negative)
  • $|a| = |-a|$ (symmetric)
  • $|ab| = |a||b|$ (multiplicative)
  • $|a + b| \leq |a| + |b|$ (triangle inequality)

AM-GM Special Cases

  • Two variables: $\frac{a + b}{2} \geq \sqrt{ab}$
  • Three variables: $\frac{a + b + c}{3} \geq \sqrt[3]{abc}$
  • Equality when all variables are equal

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