Two-Step Inequalities: Solve and Graph

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.

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

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.

A number line from −6 to 6 showing x ≥ −4: a closed circle at −4 with shading to the right. — The solution of −3x + 5 ≤ 17 is x ≥ −4: a closed circle at −4, shaded right.

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 $2x + 3 \lt 11$ .

2x + 3 - 3 \lt 11 - 3

Step 1. Subtract 3 from both sides.

2x \lt 8

Step 2. Simplify.

\frac{2x}{2} \lt \frac{8}{2}

Step 3. Divide both sides by 2. Two is positive, so don’t flip (see one-step inequalities).

x \lt 4

Step 4. Simplify.

Step 5. Graph it: open circle at 4, shade left.

Answer: $x \lt 4$

Check: Test 0: $2(0) + 3 = 3$ , and $3 \lt 11$ . True.

Example 2: Division

Solve $\frac{y}{3} - 4 \ge 1$ .

\frac{y}{3} - 4 + 4 \ge 1 + 4

Step 1. Add 4 to both sides.

\frac{y}{3} \ge 5

Step 2. Simplify.

3 \cdot \frac{y}{3} \ge 3 \cdot 5

Step 3. Multiply both sides by 3. Don’t flip, because 3 is positive.

y \ge 15

Step 4. Simplify.

Step 5. Graph it: closed circle at 15, shade right.

Answer: $y \ge 15$

Check: Test 18: $\frac{18}{3} - 4 = 6 - 4 = 2$ , and $2 \ge 1$ . True.

Example 3: Dividing by a negative

Solve $-3x + 5 \le 17$ .

-3x + 5 - 5 \le 17 - 5

Step 1. Subtract 5 from both sides. The sign stays the same.

-3x \le 12

Step 2. Simplify.

\frac{-3x}{-3} \ge \frac{12}{-3}

Step 3. Divide both sides by $-3$ . It’s negative, so flip less than or equal to into greater than or equal to.

x \ge -4

Step 4. Simplify. (See multiplying and dividing integers.)

Step 5. Graph it: closed circle at $-4$ , shade right.

Answer: $x \ge -4$

Check: Test 0: $-3(0) + 5 = 5$ , and $5 \le 17$ . True. Test $-5$ , which is outside the answer: $-3(-5) + 5 = 15 + 5 = 20$ , and 20 is not less than or equal to 17. Good.

Example 4: A negative fraction

Solve $7 - \frac{n}{2} \gt 10$ .

7 - \frac{n}{2} - 7 \gt 10 - 7

Step 1. Subtract 7 from both sides.

-\frac{n}{2} \gt 3

Step 2. Simplify.

-2 \cdot \left(-\frac{n}{2}\right) \lt -2 \cdot 3

Step 3. The variable is divided by $-2$ , so multiply both sides by $-2$ and flip the sign.

n \lt -6

Step 4. Simplify.

Step 5. Graph it: open circle at $-6$ , shade left.

Answer: $n \lt -6$

Check: Test $-8$ : $7 - \frac{-8}{2} = 7 + 4 = 11$ , and $11 \gt 10$ . True.

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?

2.5m + 3 \le 28

Step 1. Let $m$ = the number of miles. The cost must be at most $28.

2.5m + 3 - 3 \le 28 - 3

Step 2. Subtract 3 from both sides.

2.5m \le 25

Step 3. Simplify.

\frac{2.5m}{2.5} \le \frac{25}{2.5}

Step 4. Divide both sides by 2.5 (see operations with decimals).

m \le 10

Step 5. Simplify.

Answer: You can ride at most 10 miles.

Check: 10 miles cost $2.5(10) + 3 = 25 + 3 = 28$ dollars, which is exactly $28.

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

Add 4 to both sides.

x \gt 5

Divide both sides by 3.

Graph: open circle at 5, shade right

5a \le -15

Subtract 2 from both sides.

a \le -3

Divide both sides by 5.

Graph: closed circle at $-3$ , shade left

\frac{m}{4} \lt 3

Subtract 6 from both sides.

m \lt 12

Multiply both sides by 4.

Graph: open circle at 12, shade left

-2y \ge 8

Subtract 7 from both sides.

y \le -4

Divide both sides by −2 and flip the sign.

Graph: closed circle at $-4$ , shade left

-3k \gt -9

Subtract 10 from both sides.

k \lt 3

Divide both sides by −3 and flip the sign.

Graph: open circle at 3, shade left

\frac{p}{-5} \le 4

Add 1 to both sides.

p \ge -20

Multiply both sides by −5 and flip the sign.

Graph: closed circle at $-20$ , shade right

10 \lt 2w

Add 6 to both sides.

5 \lt w

Divide both sides by 2.

w \gt 5

Write the variable first.

Graph: open circle at 5, shade right

0.5x \ge 1.5

Subtract 2.5 from both sides.

x \ge 3

Divide both sides by 0.5.

Graph: closed circle at 3, shade right

-\frac{2}{3}t \lt 4

Subtract 1 from both sides.

t \gt -6

Multiply both sides by $-\frac{3}{2}$ , the reciprocal, and flip the sign (see multiplying and dividing fractions)

Graph: open circle at $-6$ , shade right

10.

12h + 42 \ge 150

Let $h$ = the number of hours. You need at least $150.

12h \ge 108

Subtract 42 from both sides.

h \ge 9

Divide both sides by 12.

You need to work at least 9 hours.

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.

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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