Solving Inequalities

Solving inequalities is similar to solving equations, with one important exception!

Same Rules Apply

You can:

• Add/subtract the same number from both sides
• Multiply/divide both sides by the same positive number

Example 1: $x + 4 < 9$

Subtract 4 from both sides: $x + 4 - 4 < 9 - 4$ $x < 5$

Example 2: $3x \geq 15$

Divide both sides by 3: $\frac{3x}{3} \geq \frac{15}{3}$ $x \geq 5$

⚠️ The Critical Rule

When you multiply or divide by a NEGATIVE number, FLIP the inequality sign!

Why? Consider: $2 < 4$

Multiply both sides by $-1$: $-2$ and $-4$

Is $-2 < -4$? No! $-2 > -4$

Example 3: $-2x > 6$

Divide by $-2$ AND flip the sign: $\frac{-2x}{-2} < \frac{6}{-2}$ $x < -3$

Example 4: $-\frac{x}{3} \leq 4$

Multiply by $-3$ AND flip the sign: $(-3) \cdot (-\frac{x}{3}) \geq (-3) \cdot 4$ $x \geq -12$

Two-Step Inequalities

Example 5: $2x - 5 > 3$

Step 1: Add 5 to both sides $2x - 5 + 5 > 3 + 5$ $2x > 8$

Step 2: Divide by 2 (positive, so no flip) $x > 4$

Example 6: $-4x + 7 \leq 19$

Step 1: Subtract 7 from both sides $-4x \leq 12$

Step 2: Divide by $-4$ AND flip the sign $x \geq -3$