Two-Step Inequalities
Two-step inequalities, like 2x + 3 \lt 11, are solved like two-step equations: undo adding or subtracting first, then undo multiplying or dividing. If you multiply or divide by a negative number, flip the inequality sign. Here, 2x \lt 8, so x \lt 4. Then graph the answer on a number line.
Before this lesson: You should know how to solve one-step inequalities and two-step equations.
What Is a Two-Step Inequality?
A two-step inequality has two operations on the variable, just like a two-step equation. In -3x + 5 \le 17, the variable is multiplied by -3, and then 5 is added.
You undo the steps in reverse order, the same way you do for equations. The only difference is the flip rule from one-step inequalities. Watch the step where you multiply or divide.
The answer to -3x + 5 \le 17 is x \ge -4, as you’ll see in Example 3. Every number from -4 up makes the inequality true.
How to Solve Two-Step Inequalities
- Undo adding or subtracting on both sides. The sign stays the same.
- Undo multiplying or dividing on both sides. If that number is negative, flip the sign.
- Graph the answer on a number line.
- Test a number from the shaded part in the original inequality.
The flip can only happen in step 2, never in step 1. Adding or subtracting a negative number doesn’t flip the sign.
Worked Examples
Example 1: A basic two-step inequality
Solve 2x + 3 \lt 11.
Step 1. Subtract 3 from both sides.
Step 2. Divide both sides by 2. Two is positive, so don’t flip.
\begin{aligned}
2x + 3 &\lt 11 \\
2x &\lt 8 \\
x &\lt 4
\end{aligned}Check: Test 0: 2(0) + 3 = 3, and 3 \lt 11. True.
Answer: x \lt 4. Open circle at 4, shade left.
Example 2: Division
Solve \frac{y}{3} - 4 \ge 1.
Step 1. Add 4 to both sides.
Step 2. Multiply both sides by 3. Don’t flip.
\begin{aligned}
\frac{y}{3} - 4 &\ge 1 \\
\frac{y}{3} &\ge 5 \\
y &\ge 15
\end{aligned}Answer: y \ge 15. Closed circle at 15, shade right.
Example 3: Dividing by a negative
Solve -3x + 5 \le 17.
Step 1. Subtract 5 from both sides. The sign stays the same.
Step 2. Divide both sides by -3. It’s negative, so flip less than or equal to into greater than or equal to.
\begin{aligned}
-3x + 5 &\le 17 \\
-3x &\le 12 \\
x &\ge -4
\end{aligned}Check: Test 0: -3(0) + 5 = 5, and 5 \le 17. True. Test -5, which is outside the answer: -3(-5) + 5 = 20, and 20 is not less than or equal to 17. Good.
Answer: x \ge -4
Example 4: A negative fraction
Solve 7 - \frac{n}{2} \gt 10.
Step 1. Subtract 7 from both sides.
Step 2. The variable is divided by -2, so multiply both sides by -2 and flip the sign.
\begin{aligned}
7 - \frac{n}{2} &\gt 10 \\
-\frac{n}{2} &\gt 3 \\
n &\lt -6
\end{aligned}Answer: n \lt -6. Open circle at -6, shade left.
Example 5: Word problem
A taxi charges $3 to start plus $2.50 for each mile. You have $28. How many miles can you ride?
Step 1. Let m = the number of miles. The cost must be at most $28.
2.5m + 3 \le 28
Step 2. Subtract 3, then divide by 2.5.
\begin{aligned}
2.5m &\le 25 \\
m &\le 10
\end{aligned}Answer: You can ride at most 10 miles.
Common Mistakes
| Mistake | What it looks like | How to fix it |
|---|---|---|
| Flipping when you add or subtract | -3x + 5 \le 17 becomes -3x \ge 12 | Subtracting 5 never flips the sign. Only flip when you divide by the negative. |
| Forgetting to flip | -3x \le 12 becomes x \le -4 | You divided by -3, so flip: x \ge -4. |
| Flipping because a number is negative | x - 4 \gt -6 becomes x \lt -2 | You added 4. Nothing was multiplied or divided, so x \gt -2. |
| Dividing only one term | 2x + 3 \lt 11 becomes x + 3 \lt 5.5 | Undo the adding first, then divide. |
| Using the wrong circle | Graphing x \ge -4 with an open circle | Greater than or equal to includes -4, so use a closed circle. |
When the Variable Is on the Right
Sometimes the variable ends up on the right side. Solve as usual, then read the answer carefully. For 9 \gt 4x + 1:
\begin{aligned}
9 &\gt 4x + 1 \\
8 &\gt 4x \\
2 &\gt x
\end{aligned}2 \gt x means the same thing as x \lt 2. Rewrite it with the variable first before you graph, so you shade the right way. Longer inequalities with parentheses are covered in multi-step inequalities.
Practice Problems
Solve each inequality. Then describe its graph.
- 3x - 4 \gt 11
- 5a + 2 \le -13
- \frac{m}{4} + 6 \lt 9
- -2y + 7 \ge 15
- 10 - 3k \gt 1
- \frac{p}{-5} - 1 \le 3
- 4 \lt 2w - 6
- 0.5x + 2.5 \ge 4
- -\frac{2}{3}t + 1 \lt 5
- You need at least $150 for a trip. You already have $42, and you earn $12 an hour. How many hours do you need to work?
Answers
Show answers
- 3x \gt 15, so x \gt 5. Open circle at 5, shade right.
- 5a \le -15, so a \le -3. Closed circle at -3, shade left.
- \frac{m}{4} \lt 3, so m \lt 12. Open circle at 12, shade left.
- -2y \ge 8. Divide by -2 and flip: y \le -4. Closed circle at -4, shade left.
- -3k \gt -9. Divide by -3 and flip: k \lt 3. Open circle at 3, shade left.
- \frac{p}{-5} \le 4. Multiply by -5 and flip: p \ge -20. Closed circle at -20, shade right.
- 10 \lt 2w, so 5 \lt w, which is w \gt 5. Open circle at 5, shade right.
- 0.5x \ge 1.5, so x \ge 3. Closed circle at 3, shade right.
- -\frac{2}{3}t \lt 4. Multiply by -\frac{3}{2} and flip: t \gt -6. Open circle at -6, shade right.
- 12h + 42 \ge 150, so 12h \ge 108 and h \ge 9. You need to work at least 9 hours.
Two-Step Inequalities FAQ
What is a two-step inequality?
It is an inequality that takes two inverse operations to solve, such as 4x - 1 \ge 7.
How do you solve two-step inequalities?
Undo adding or subtracting first, then undo multiplying or dividing. Flip the sign if you multiply or divide by a negative number.
When do you flip the sign in a two-step inequality?
Only in the multiply-or-divide step, and only when that number is negative. The adding or subtracting step never flips the sign.
How do you check the answer to an inequality?
Pick a number from your answer and put it in the original inequality. It should be true. A number outside your answer should make it false.
Summary
- Solve two-step inequalities in the same order as two-step equations.
- Undo adding or subtracting first. The sign doesn’t change.
- Undo multiplying or dividing next. Flip the sign if the number is negative.
- Rewrite an answer like 2 \gt x as x \lt 2 before graphing.
- Test a number from your answer in the original inequality.
Previous lesson: One-Step Inequalities
Next lesson: Multi-Step Inequalities
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