5. Hardened Inequalities

Inequalities follow equation rules with one extra rule: multiplying or dividing by a negative flips the inequality sign.

Example 1:
\[ -3(x-2)\le 9. \]
Distribute:
\[ -3x+6\le 9. \]
Subtract 6:
\[ -3x\le 3. \]
Divide by \(-3\) and flip:
\[ x\ge -1. \]

Example 2 - variables on both sides:
\[ 5-2x>3x-15. \]
Add \(2x\):
\[ 5>5x-15. \]
Add 15:
\[ 20>5x,\qquad 4>x,\qquad x<4. \]

Example 3 - compound:
\[ -4< 2x-6 \le 8. \]
Add 6 to all three parts:
\[ 2< 2x \le 14. \]
Divide all parts by 2:
\[ 1< x \le 7. \]

Common mistake: forgetting to flip the sign only when you divide by a negative. Adding or subtracting never flips it.