Solving Inequalities Worksheet

Q: How do you solve an inequality?

Solve it like an equation: undo addition or subtraction, then multiplication or division, doing the same thing to both sides. The one difference is that when you multiply or divide both sides by a negative number, you must flip the inequality sign.

Q: Why do you flip the sign when dividing by a negative?

Multiplying by a negative reverses the order of numbers on the number line. For example, 2 -5. So if -3x -4, not x -4.

25 printable problems: one-step and multi-step inequalities, compound inequalities, word problems and graphing inequalities in two variables. The answer key shows interval notation and the number-line graph for every problem.

25 problems · answer key included · free, no sign-up

Solving Inequalities Worksheet

learnappu.com/math/algebra/linear-inequalities/worksheet

Name ______________________Date __________ Period ____

Teachers and parents: print the student copy for class or homework, or print with the answer key on a separate page. You can also link to this page from your course site or LMS; no account is needed. Problems get harder within each section.

A. One-step inequalities

Solve and graph on the number line. Flip the sign when you multiply or divide by a negative.

1
$x + 4 > 9$
2
$y - 3 \leq - 7$
3
$5 n \geq 30$
4
$- 3 x < 12$
5
$\frac{k}{- 2} \geq 3$
6
$\frac{x}{4} < - 1$

B. Two-step and multi-step inequalities

Solve, write the answer in interval notation, and graph.

7
$2 x + 3 < 11$
8
$- 5 x + 2 \geq 17$
9
$3 (x - 1) > 9$
10
$7 - x \leq 10$
11
$\frac{x}{3} - 2 > 1$
12
$4 x - 5 \leq 2 x + 7$
13
$- 2 (x + 4) < 6$
14
$6 - 3 x \geq 2 x - 9$

C. Compound inequalities

Solve. "And" means both are true (overlap); "or" means either is true.

15
$- 3 < x + 2 < 7$
16
$1 \leq 2 x - 3 \leq 9$
17
$3 x < - 6 or x - 1 > 2$
18
$- 4 \leq - 2 x < 6$

D. Word problems

Write an inequality, solve it, and answer the question in a sentence.

19
You have $50. After buying an $8 backpack, you want notebooks that cost $3.50 each. What is the greatest number of notebooks you can buy?
20
You scored 80 and 88 on two tests. What score do you need on the third test to average at least 85?
21
A phone plan costs $20 a month plus $0.10 per text. How many texts can you send and keep the bill under $35?

E. Graph inequalities in two variables

Graph the boundary line (dashed for < or >, solid for ≤ or ≥), then shade the side that makes the inequality true.

22
$y > x + 1$
23
$y \leq - 2 x + 4$
24
$y \geq \frac{1}{2} x - 3$
25
$2 x + y < 6$

Answer key

Check your work after you finish each section.

A. One-step inequalities

1.
$x > 5$ · $(5, \infty)$
Subtract 4
2.
$y \leq - 4$ · $(- \infty, - 4]$
Add 3
3.
$n \geq 6$ · $[6, \infty)$
Divide by 5
4.
$x > - 4$ · $(- 4, \infty)$
Divide by $- 3$ and flip the sign
5.
$k \leq - 6$ · $(- \infty, - 6]$
Multiply by $- 2$ and flip the sign
6.
$x < - 4$ · $(- \infty, - 4)$
Multiply by 4

B. Two-step and multi-step inequalities

7.
$x < 4$ · $(- \infty, 4)$
Subtract 3, divide by 2
8.
$x \leq - 3$ · $(- \infty, - 3]$
Subtract 2, divide by $- 5$ , flip
9.
$x > 4$ · $(4, \infty)$
Distribute or divide by 3: $x - 1 > 3$
10.
$x \geq - 3$ · $[- 3, \infty)$
Subtract 7: $- x \leq 3$ , then flip
11.
$x > 9$ · $(9, \infty)$
Add 2, multiply by 3
12.
$x \leq 6$ · $(- \infty, 6]$
Variables to one side: $2 x \leq 12$
13.
$x > - 7$ · $(- 7, \infty)$
$- 2 x - 8 < 6$ , so $- 2 x < 14$ ; flip
14.
$x \leq 3$ · $(- \infty, 3]$
$15 \geq 5 x$

C. Compound inequalities

15.
$- 5 < x < 5$ · $(- 5, 5)$
Subtract 2 from all three parts
16.
$2 \leq x \leq 6$ · $[2, 6]$
Add 3, then divide all parts by 2
17.
$x < - 2 or x > 3$ · $(- \infty, - 2) \cup (3, \infty)$
Solve each part separately
18.
$- 3 < x \leq 2$ · $(- 3, 2]$
Divide all parts by $- 2$ and flip both signs

D. Word problems

19.
$n \leq 12$ , so at most $12$ notebooks
Inequality: $8 + 3.5 n \leq 50$
20.
$t \geq 87$
Inequality: $\frac{80 + 88 + t}{3} \geq 85$
21.
$s < 150$ , so at most $149$ texts
Inequality: $20 + 0.10 s < 35$

E. Graph inequalities in two variables

22.
Dashed line $y = x + 1$ (points on it are not solutions), shade above. Test point $(0, 0)$ is not a solution.
23.
Solid line $y = - 2 x + 4$ , shade below. Test point $(0, 0)$ is a solution.
24.
Solid line $y = \frac{1}{2} x - 3$ , shade above. Test point $(0, 0)$ is a solution.
25.
Rewrite as $y < - 2 x + 6$ . Dashed line, shade below. Test point $(0, 0)$ is a solution.

How to solve and graph inequalities

Solve it like an equation. For $2 x + 3 < 11$ , subtract 3 and divide by 2 to get $x < 4$ .

Flip the sign for negatives. Multiplying or dividing both sides by a negative number reverses the inequality: $- 3 x < 12$ becomes $x > - 4$ .

Graph it. Open circle for $<$ or $>$ , closed circle for $\leq$ or $\geq$ , then shade toward the solutions.

Write interval notation. $x < 4$ is $(- \infty, 4)$ ; $x \geq 6$ is $[6, \infty)$ . Infinity always gets a parenthesis.

Check with a number from your answer: $x = 0$ in $2 x + 3 < 11$ gives $3 < 11$ . True.

Read the full Linear Inequalities lesson with worked examples →

Frequently asked questions

How do you solve an inequality?

Solve it like an equation: undo addition or subtraction, then multiplication or division, doing the same thing to both sides. The one difference is that when you multiply or divide both sides by a negative number, you must flip the inequality sign.

Why do you flip the sign when dividing by a negative?

Multiplying by a negative reverses the order of numbers on the number line. For example, 2 < 5, but -2 > -5. So if -3x < 12, dividing by -3 gives x > -4, not x < -4. Check with a number: x = 0 makes -3(0) < 12 true, and 0 > -4.

Open or closed circle on the number line?

Use an open circle for < and > because the endpoint is not a solution. Use a closed (filled) circle for ≤ and ≥ because the endpoint is included. In interval notation, open circles become parentheses ( ) and closed circles become brackets [ ].

What is a compound inequality?

Two inequalities joined by "and" or "or". An "and" inequality like -3 < x + 2 < 7 is true only where both parts overlap, giving one segment. An "or" inequality like x < -2 or x > 3 is true if either part is true, giving two separate rays.

How do you graph an inequality like y > x + 1?

Graph the boundary line y = x + 1, dashed for < or > and solid for ≤ or ≥. Then pick a test point not on the line, usually (0, 0). If it makes the inequality true, shade its side; if not, shade the other side.

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