An inequality like $3x - 4 > 11$ has many solutions. You can solve it almost exactly like a two-step equation, and graph the answer the way you graphed inequalities in Grade 6. There is one new rule.
Key idea
Solve the way you would solve an equation. But when you multiply or divide both sides by a negative number, reverse the inequality sign ($<$ becomes $>$, $\le$ becomes $\ge$).
Why it works
Adding or subtracting slides both sides the same way, so their order stays the same. Multiplying by a negative number flips the number line around $0$: $2 < 5$, but $-2 > -5$, because $-2$ is to the right of $-5$. So after multiplying or dividing by a negative, the bigger side becomes the smaller side, and the sign must turn around to stay true.
Worked examples
Worked example: No flip needed
Solve $3x - 4 > 11$.
- Add 4 to both sides: $3x > 15$.
- Divide by 3 (positive, so the sign stays): $x > 5$.
- Check $x = 6$: $3(6) - 4 = 14 > 11$ ✓.
Answer: $x > 5$
Worked example: Dividing by a negative
Solve $-2x + 7 \le 1$.
- Subtract 7: $-2x \le -6$.
- Divide by $-2$ and reverse the sign: $x \ge 3$.
- Check $x = 4$: $-8 + 7 = -1 \le 1$ ✓. Check $x = 0$ (not in the answer): $7 \le 1$ is false ✓.
Answer: $x \ge 3$
Worked example: A word problem
You have \$40. A fair ticket costs \$12 and each ride costs \$3.50. How many rides can you take?
- Let $r$ be the number of rides: $12 + 3.5r \le 40$ ("at most \$40").
- Subtract 12: $3.5r \le 28$. Divide by 3.5: $r \le 8$.
- You can take at most 8 rides.
Answer: At most 8 rides
Common mistake
Forgetting to reverse the sign after dividing by a negative: $-2x \le -6$ becoming $x \le 3$. Test a number from your answer: $x = 0$ gives $7 \le 1$, which is false, so $x \le 3$ cannot be right. Testing one number catches this every time.