One-Step Inequalities
A one-step inequality, like x + 4 \gt 9, is solved like a one-step equation: do the inverse operation to both sides. There is one new rule for one-step inequalities. If you multiply or divide both sides by a negative number, flip the inequality sign. The answer, x \gt 5, is a range of numbers you can graph on a number line.
Before this lesson: You should know how to solve one-step equations and how to compare negative numbers on a number line.
What Is an Inequality?
An equation says two things are equal. An inequality says one side is bigger or smaller than the other. It uses one of these four signs:
| Sign | Meaning | Example | On a number line |
|---|---|---|---|
| \lt | is less than | x \lt 3 | open circle at 3, shade left |
| \gt | is greater than | x \gt 3 | open circle at 3, shade right |
| \le | is less than or equal to | x \le 3 | closed circle at 3, shade left |
| \ge | is greater than or equal to | x \ge 3 | closed circle at 3, shade right |
An equation like x + 4 = 9 has one answer. An inequality like x + 4 \gt 9 has many answers: 6, 7, 10.5 and every other number greater than 5. So the answer is written as an inequality and drawn on a number line.
An open circle means the number itself is not part of the answer. A closed circle means it is. For x \gt 2, 2 is not included, but for x \le -1, -1 is.
How to Solve One-Step Inequalities
Solve it like an equation, using the inverse operation on both sides. Then check the one special rule.
| What you do to both sides | What happens to the sign |
|---|---|
| Add or subtract any number | It stays the same. |
| Multiply or divide by a positive number | It stays the same. |
| Multiply or divide by a negative number | Flip it. Less than becomes greater than, and greater than becomes less than. |
Why flip? Start with a true statement, 2 \lt 5. Multiply both sides by -1 and you get -2 and -5. On the number line, -2 is to the right of -5, so now -2 \gt -5. Multiplying by a negative reverses the order, so the sign must flip.
Worked Examples
Example 1: Addition
Solve x + 4 \gt 9 and graph it.
Step 1. Subtract 4 from both sides. Subtracting doesn’t change the sign.
\begin{aligned}
x + 4 &\gt 9 \\
x &\gt 5
\end{aligned}Step 2. Graph it: open circle at 5, shade to the right.
Check: Try a number in the answer, like 6: 6 + 4 = 10, and 10 \gt 9. True.
Answer: x \gt 5
Example 2: Subtraction
Solve y - 3 \le -1.
Step 1. Add 3 to both sides.
\begin{aligned}
y - 3 &\le -1 \\
y &\le 2
\end{aligned}Step 2. Graph it: closed circle at 2, shade to the left.
Answer: y \le 2
Example 3: Dividing by a positive number
Solve 6n \ge -18.
Step 1. Divide both sides by 6. Six is positive, so the sign stays the same, even though -18 is negative.
\begin{aligned}
6n &\ge -18 \\
n &\ge -3
\end{aligned}Answer: n \ge -3. Only the number you multiply or divide by decides whether to flip.
Example 4: Dividing by a negative number
Solve -4x \lt 20.
Step 1. Divide both sides by -4. Because you divide by a negative, flip less than to greater than.
\begin{aligned}
-4x &\lt 20 \\
\frac{-4x}{-4} &\gt \frac{20}{-4} \\
x &\gt -5
\end{aligned}Check: Try 0, which is greater than -5: -4(0) = 0, and 0 \lt 20. True.
Answer: x \gt -5
Example 5: Word problem
Ana has $40 to spend on T-shirts that cost $12 each. How many T-shirts can she buy?
Step 1. Let s = the number of shirts. The cost can’t be more than $40, so the cost is less than or equal to 40.
12s \le 40
Step 2. Divide both sides by 12.
s \le \frac{40}{12} \approx 3.33Step 3. She can’t buy part of a shirt, so the answer must be a whole number.
Answer: Ana can buy at most 3 T-shirts. Check: 3 shirts cost $36, but 4 would cost $48.
Common Mistakes
| Mistake | What it looks like | How to fix it |
|---|---|---|
| Flipping because the answer is negative | 6n \ge -18 becomes n \le -3 | You divided by positive 6, so don’t flip: n \ge -3. |
| Forgetting to flip | -4x \lt 20 becomes x \lt -5 | You divided by -4, so flip: x \gt -5. |
| Using the wrong circle | Graphing x \le 2 with an open circle | Use a closed circle for less than or equal to, and greater than or equal to. |
| Shading the wrong way | Graphing 7 \lt x by shading left | Rewrite it as x \gt 7, then shade right. |
| Writing just one number | Answering x = 5 | An inequality has many answers. Write x \gt 5. |
Quick check: Pick a number in your shaded region and test it in the original inequality. It should make a true statement.
Writing an Inequality from Words
Word problems use phrases instead of signs. Here are the common ones:
| Phrase | Sign | Example |
|---|---|---|
| at most, no more than | \le | at most 30 people: p \le 30 |
| at least, no less than | \ge | at least 18 years old: a \ge 18 |
| more than, over | \gt | more than 5 laps: n \gt 5 |
| fewer than, under | \lt | under 2 hours: t \lt 2 |
Practice Problems
Solve each inequality. Then describe its graph.
- x + 7 \lt 12
- a - 5 \ge -2
- -3 \gt y + 4
- 8k \le 56
- -5p \gt 35
- \frac{n}{3} \ge -4
- \frac{w}{-6} \lt 2
- -x \le 9
- 0.5t \gt 3
- An elevator can carry at most 1,800 pounds. How many 60-pound boxes can it carry?
Answers
Show answers
- x \lt 5. Open circle at 5, shade left.
- a \ge 3. Closed circle at 3, shade right.
- Subtract 4: -7 \gt y, which is y \lt -7. Open circle at -7, shade left.
- k \le 7. Closed circle at 7, shade left.
- Divide by -5 and flip: p \lt -7. Open circle at -7, shade left.
- Multiply by 3: n \ge -12. Closed circle at -12, shade right.
- Multiply by -6 and flip: w \gt -12. Open circle at -12, shade right.
- Divide by -1 and flip: x \ge -9. Closed circle at -9, shade right.
- Divide by 0.5: t \gt 6. Open circle at 6, shade right.
- 60b \le 1800, so b \le 30. It can carry at most 30 boxes.
One-Step Inequalities FAQ
What is a one-step inequality?
It is an inequality you can solve with one inverse operation, such as x - 2 \lt 6 or 3x \ge 12.
When do you flip the inequality sign?
Only when you multiply or divide both sides by a negative number. Adding, subtracting, or using a positive number never flips it.
How do you graph an inequality on a number line?
Put a circle on the boundary number: open for less than or greater than, closed for or equal to. Then shade the side that makes the inequality true.
What is the difference between less than and less than or equal to?
x \lt 3 does not include 3, but x \le 3 does. That’s why one uses an open circle and the other a closed circle.
Summary
- Solve a one-step inequality with the same inverse operation you would use for an equation.
- Flip the sign only when you multiply or divide by a negative number.
- The answer is a range of numbers, written as an inequality.
- Graph with an open circle for less than or greater than, and a closed circle when the number is included.
- Test a number from your answer in the original inequality.
Previous lesson: Literal Equations
Next lesson: Two-Step Inequalities
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