You have solved two-step inequalities, and you know that every equation step must keep the balance. Inequalities with more steps use the same moves, with one rule to watch: multiplying or dividing by a negative number reverses the inequality.
Why it works
Adding or subtracting slides every number the same distance along the number line, so their order never changes: if $2 < 5$, then $2 - 7 < 5 - 7$. Multiplying by a negative number is different. Multiplying by $-1$ reflects every number across $0$: $2$ goes to $-2$ and $5$ goes to $-5$. Now the order is backward, because $-2 > -5$. Every negative multiplier includes that reflection, so after multiplying or dividing both sides by a negative number, the sign must flip to stay true.
Key idea
Solve an inequality like an equation. Whenever you multiply or divide both sides by a negative number, reverse the direction of the sign. Adding and subtracting, even negative numbers, never flips it.
Worked examples
Worked example: Dividing by a negative
Solve $-2x - 4 \ge 6$.
- Add $4$ to both sides: $-2x \ge 10$.
- Divide both sides by $-2$ and flip the sign: $x \le -5$.
- Check $x = -6$: $-2(-6) - 4 = 8$, and $8 \ge 6$ is true. Check $x = 0$: $-4 \ge 6$ is false, as it should be.
Answer: $x \le -5$
Worked example: Two ways to finish
Solve $3(x - 4) < 5x + 2$.
- Distribute: $3x - 12 < 5x + 2$.
- Way 1: subtract $5x$: $-2x - 12 < 2$. Add $12$: $-2x < 14$. Divide by $-2$ and flip: $x > -7$.
- Way 2: subtract $3x$ instead: $-12 < 2x + 2$. Subtract $2$: $-14 < 2x$. Divide by $2$ (no flip): $-7 < x$.
- Both say the same thing: $x$ is greater than $-7$.
Answer: $x > -7$
Common mistake
Two opposite slips: forgetting to flip after dividing by a negative, and flipping when you did not. In $x - (-3) > 1$, you subtract $3$ from both sides, which is not multiplying by a negative, so no flip: $x > -2$. Check by testing one number from your answer in the original inequality; if it fails, look for a sign slip.