One-Step Inequalities: Solve and Graph

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.

Extra practice: The Algebra 1 Unit 2 Practice Pack has printable problems, answer keys and a cheat sheet for this unit.

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.

Two number lines from −6 to 6. The first graphs x > 2 with an open circle at 2 and shading to the right. The second graphs x ≤ −1 with a closed circle at −1 and shading to the left. — Open circle: the number is not included. Closed circle: it is included.

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.

x + 4 - 4 \gt 9 - 4

Step 1. Subtract 4 from both sides. Subtracting doesn’t change the sign.

x \gt 5

Step 2. Simplify each side.

Step 3. Graph it: open circle at 5, shade to the right.

Answer: $x \gt 5$

Check: Try a number in the answer, like 6: $6 + 4 = 10$ , and $10 \gt 9$ . True.

Example 2: Subtraction

Solve $y - 3 \le -1$ .

y - 3 + 3 \le -1 + 3

Step 1. Add 3 to both sides.

y \le 2

Step 2. Simplify each side. $-1 + 3 = 2$ . (See adding and subtracting integers.)

Step 3. Graph it: closed circle at 2, shade to the left.

Answer: $y \le 2$

Check: Try 0: $0 - 3 = -3$ , and $-3 \le -1$ . True.

Example 3: Dividing by a positive number

Solve $6n \ge -18$ .

\frac{6n}{6} \ge \frac{-18}{6}

Step 1. Divide both sides by 6. Six is positive, so the sign stays the same, even though $-18$ is negative.

n \ge -3

Step 2. Simplify each side. (See multiplying and dividing integers.)

Step 3. Graph it: closed circle at $-3$ , shade to the right.

Answer: $n \ge -3$ . Only the number you multiply or divide by decides whether to flip.

Example 4: Dividing by a negative number

Solve $-4x \lt 20$ .

\frac{-4x}{-4} \gt \frac{20}{-4}

Step 1. Divide both sides by $-4$ . Because you divide by a negative, flip less than to greater than.

x \gt -5

Step 2. Simplify each side.

Step 3. Graph it: open circle at $-5$ , shade to the right.

Answer: $x \gt -5$

Check: Try 0, which is greater than $-5$ : $-4(0) = 0$ , and $0 \lt 20$ . True.

Example 5: Word problem

Ana has $40 to spend on T-shirts that cost $12 each. How many T-shirts can she buy?

12s \le 40

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.

\frac{12s}{12} \le \frac{40}{12}

Step 2. Divide both sides by 12.

s \le \frac{10}{3}

Step 3. Simplify. $\frac{40}{12} = \frac{10}{3}$ , which is about 3.33.

Step 4. She can’t buy part of a shirt, so the answer must be a whole number. The largest whole number that works is 3.

Answer: Ana can buy at most 3 T-shirts.

Check: 3 shirts cost $12(3) = 36$ dollars, but 4 would cost $12(4) = 48$ dollars, which is more than $40.

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 + 7 - 7 \lt 12 - 7

Subtract 7 from both sides.

x \lt 5

Simplify.

Graph: open circle at 5, shade left

a - 5 + 5 \ge -2 + 5

Add 5 to both sides.

a \ge 3

Simplify.

Graph: closed circle at 3, shade right

-3 - 4 \gt y + 4 - 4

Subtract 4 from both sides.

-7 \gt y

Simplify.

y \lt -7

Write the variable first. The sign turns around with it.

Graph: open circle at $-7$ , shade left

\frac{8k}{8} \le \frac{56}{8}

Divide both sides by 8.

k \le 7

Simplify.

Graph: closed circle at 7, shade left

\frac{-5p}{-5} \lt \frac{35}{-5}

Divide both sides by −5 and flip the sign.

p \lt -7

Simplify.

Graph: open circle at $-7$ , shade left

3 \cdot \frac{n}{3} \ge 3(-4)

Multiply both sides by 3.

n \ge -12

Simplify.

Graph: closed circle at $-12$ , shade right

-6 \cdot \frac{w}{-6} \gt -6 \cdot 2

Multiply both sides by −6 and flip the sign.

w \gt -12

Simplify.

Graph: open circle at $-12$ , shade right

\frac{-x}{-1} \ge \frac{9}{-1}

Divide both sides by −1 and flip the sign.

x \ge -9

Simplify.

Graph: closed circle at $-9$ , shade right

\frac{0.5t}{0.5} \gt \frac{3}{0.5}

Divide both sides by 0.5 (see operations with decimals)

t \gt 6

Simplify.

Graph: open circle at 6, shade right

10.

60b \le 1800

Let $b$ = the number of boxes. The weight can be at most 1,800 pounds.

\frac{60b}{60} \le \frac{1800}{60}

Divide both sides by 60.

b \le 30

Simplify.

It can carry at most 30 boxes.

Want more practice? The Algebra 1 Unit 2 Practice Pack has 51 pages of problems on this and the other lessons in the unit, with step-by-step answers and a cheat sheet.

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.

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Next lesson: Two-Step Inequalities

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