4. The Sign-Flip Rule

Here is the special rule that makes inequalities different from equations:

When you multiply or divide both sides by a negative number, flip the inequality sign.

Example 1:
\[ -3x < 12. \]
Divide by \(-3\). Because the divisor is negative, flip \(<\) to \(>\):
\[ x > -4. \]

Example 2:
\[ -2x+7\le 15. \]
Subtract 7:
\[ -2x\le 8. \]
Divide by \(-2\) and flip:
\[ x\ge -4. \]

Why does the sign flip? Try true numbers:
\[ 2<5. \]
Multiply both sides by \(-1\):
\[ -2>-5. \]
The direction must reverse to stay true.

Common mistake: flipping when subtracting a negative. The flip happens only when multiplying or dividing by a negative, not simply because a negative number appears.