📊 Introduction to Inequalities

Inequality Symbols

• $>$ means "greater than" (e.g., $5 > 3$)
• $<$ means "less than" (e.g., $2 < 7$)
• $\geq$ means "greater than or equal to" (e.g., $x \geq 4$)
• $\leq$ means "less than or equal to" (e.g., $x \leq 10$)

Real Example: "You must be at least 13 years old" → $\text{age} \geq 13$ 🎂

Graphing Rules

• Open circle ○: Use for $>$ or $<$ (number NOT included)
• Closed circle ●: Use for $\geq$ or $\leq$ (number IS included)
• Shade/Arrow: Shows all solutions in that direction

Example: $x > 3$ → Open circle at 3, shade to the right →

Example: $x \leq 5$ → Closed circle at 5, shade to the left ←

Common Phrases

• "More than" → $>$ (e.g., "More than 10" → $x > 10$)
• "Less than" → $<$ (e.g., "Less than 20" → $x < 20$)
• "At least" → $\geq$ (e.g., "At least 5" → $x \geq 5$)
• "At most" → $\leq$ (e.g., "At most 15" → $x \leq 15$)

🎮 Example: "A game requires at least 2 players" → $p \geq 2$

The Golden Rule ⚠️

When you multiply or divide by a negative number, flip the inequality sign!

Example 1: $x + 3 < 10$

Subtract 3 from both sides:

$$x < 7$$

Example 2: $-2x > 6$

Divide both sides by -2 (flip the sign!):

$$x < -3$$

Example: $-1 \leq x < 4$

This means: $x$ is greater than or equal to -1 AND less than 4

Solutions: -1, 0, 1, 2, 3 (but NOT 4)

📊 On a number line: Closed circle at -1, open circle at 4, shade between them