M
Math Coach Amy
Amy Ferguson Moncure

Algebra 1 · Unit 1

Solve Multi-Step Inequalities

Solve linear inequalities in one variable and graph the solution on a number line.

Jump to section:

Key Ideas Easy Medium Medium Graph Medium Write inequality Hard AND Hard OR Special Cases Super Challenge
What to Remember

Graph on a number line

x > 5   open circle

x ≥ 5   closed circle

Compound inequalities

AND 1 < x ≤ 4

OR x ≤ −2 or x > 3

Compound inequality special cases

x > 3 AND x > 7

x > 3x > 7

AND overlap → x > 7

x > 3 OR x > 7

x > 3x > 7

OR anything shaded works! → x > 3

x > 1 AND x < −3

x > 1x < −3

AND means overlap. No overlap → no solution

x > 1 OR x < −3

x > 1x < −3

OR anything shaded works → infinitely many solutions

Easy example

x + 4 ≤ 9

x ≤ 5

Medium example

−3x + 5 > 14

−3x > 9

x < −3 (flip the sign)

Medium Graph example

x > 5 or x ≤ 1

Medium Write inequality example

−5 < x ≤ 1

Hard AND example

−2 < 3x + 1 ≤ 10

−1 < x ≤ 3

Hard OR example

x + 2 ≤ −1 or 3x > 12

x ≤ −3 or x > 4

Level 1

Easy

Two steps, then graph.

Example
Solve and graph. x + 4 ≤ 9

x + 4 ≤ 9

x ≤ 5

Answer: x ≤ 5

1
Solve and graph. x − 3 > 2
Show Answer

x − 3 > 2

x > 5

Answer: x > 5

2
Solve and graph. 2x ≤ 10
Show Answer

2x ≤ 10

x ≤ 5

Answer: x ≤ 5

3
Solve and graph. x + 7 ≥ 1
Show Answer

x + 7 ≥ 1

x ≥ −6

Answer: x ≥ −6

4
Solve and graph. 5x < 20
Show Answer

5x < 20

x < 4

Answer: x < 4

5
Solve and graph. x + 11 < 5
Show Answer

x + 11 < 5

x < −6

Answer: x < −6

Level 2

Medium

Distribute. When you multiply or divide by a negative, flip the sign.

Example
Solve and graph. −3x + 5 > 14

−3x + 5 > 14

−3x > 9

x < −3 (flip the sign)

Answer: x < −3

1
Solve and graph. 2x − 7 ≥ 9
Show Answer

2x − 7 ≥ 9

2x ≥ 16

x ≥ 8

Answer: x ≥ 8

2
Solve and graph. 4 − x < 1
Show Answer

4 − x < 1

−x < −3

x > 3 (flip the sign)

Answer: x > 3

3
Solve and graph. 3(x + 2) ≤ 15
Show Answer

3(x + 2) ≤ 15

3x + 6 ≤ 15

3x ≤ 9

x ≤ 3

Answer: x ≤ 3

4
Solve and graph. −4x + 3 ≤ 19
Show Answer

−4x + 3 ≤ 19

−4x ≤ 16

x ≥ −4 (flip the sign)

Answer: x ≥ −4

5
Solve and graph. −2x + 6 > 0
Show Answer

−2x + 6 > 0

−2x > −6

x < 3 (flip the sign)

Answer: x < 3

Level 2

Medium Graph

Graph a compound inequality.

Example
Graph. x > 5 or x ≤ 1

x > 5 or x ≤ 1

Answer: x ≤ 1 or x > 5

1
Graph. x ≤ 6 and x ≥ −1
Show Answer

x ≤ 6 and x ≥ −1

−1 ≤ x ≤ 6

Answer: −1 ≤ x ≤ 6

2
Graph. x > 2 and x > 4
Show Answer

x > 2 and x > 4

AND overlap → x > 4

Answer: x > 4

3
Graph. x < 0 or x ≥ 3
Show Answer

x < 0 or x ≥ 3

Answer: x < 0 or x ≥ 3

4
Graph. x < 8 and x < 3
Show Answer

x < 8 and x < 3

AND overlap → x < 3

Answer: x < 3

5
Graph. x ≥ 2 or x ≤ −4
Show Answer

x ≥ 2 or x ≤ −4

Answer: x ≤ −4 or x ≥ 2

Level 2

Medium Write inequality

Write a compound inequality from a graph.

Example
Write a compound inequality for the graph.

−5 < x ≤ 1

Answer: −5 < x ≤ 1

1
Write a compound inequality for the graph.
Show Answer

0 ≤ x ≤ 4

Answer: 0 ≤ x ≤ 4

2
Write a compound inequality for the graph.
Show Answer

x ≤ −3 or x ≥ 5

Answer: x ≤ −3 or x ≥ 5

3
Write a compound inequality for the graph.
Show Answer

−3 < x < 2

Answer: −3 < x < 2

4
Write a compound inequality for the graph.
Show Answer

x < −2 or x > 4

Answer: x < −2 or x > 4

5
Write a compound inequality for the graph.
Show Answer

−6 ≤ x < 1

Answer: −6 ≤ x < 1

Level 3

Hard AND

Rewrite as two inequalities, then solve each piece.

Example
Solve and graph. −2 < 3x + 1 ≤ 10

−2 < 3x + 1 ≤ 10

−2 < 3x + 1

−3 < 3x

−1 < x

3x + 1 ≤ 10

3x ≤ 9

x ≤ 3

−1 < x ≤ 3

Answer: −1 < x ≤ 3

1
Solve and graph. 1 ≤ 2x − 3 < 7
Show Answer

1 ≤ 2x − 3 < 7

1 ≤ 2x − 3

4 ≤ 2x

2 ≤ x

2x − 3 < 7

2x < 10

x < 5

2 ≤ x < 5

Answer: 2 ≤ x < 5

2
Solve and graph. −5 < x + 1 ≤ 3
Show Answer

−5 < x + 1 ≤ 3

−5 < x + 1

−6 < x

x + 1 ≤ 3

x ≤ 2

−6 < x ≤ 2

Answer: −6 < x ≤ 2

3
Solve and graph. −1 ≤ 4x + 3 < 11
Show Answer

−1 ≤ 4x + 3 < 11

−1 ≤ 4x + 3

−4 ≤ 4x

−1 ≤ x

4x + 3 < 11

4x < 8

x < 2

−1 ≤ x < 2

Answer: −1 ≤ x < 2

4
Solve and graph. 2 < 2x − 4 ≤ 8
Show Answer

2 < 2x − 4 ≤ 8

2 < 2x − 4

6 < 2x

3 < x

2x − 4 ≤ 8

2x ≤ 12

x ≤ 6

3 < x ≤ 6

Answer: 3 < x ≤ 6

5
Solve and graph. −4 < 2(x − 1) ≤ 6
Show Answer

−4 < 2(x − 1) ≤ 6

−4 < 2(x − 1)

−2 < x − 1

−1 < x

2(x − 1) ≤ 6

x − 1 ≤ 3

x ≤ 4

−1 < x ≤ 4

Answer: −1 < x ≤ 4

Level 3

Hard OR

Solve each piece, then graph both rays.

Example
Solve and graph. x + 2 ≤ −1 or 3x > 12

x + 2 ≤ −1 or 3x > 12

x + 2 ≤ −1

x ≤ −3

3x > 12

x > 4

x ≤ −3 or x > 4

Answer: x ≤ −3 or x > 4

1
Solve and graph. x − 5 ≥ 1 or 2x < −8
Show Answer

x − 5 ≥ 1 or 2x < −8

x − 5 ≥ 1

x ≥ 6

2x < −8

x < −4

x < −4 or x ≥ 6

Answer: x < −4 or x ≥ 6

2
Solve and graph. 4x + 1 ≤ 9 or x − 3 > 2
Show Answer

4x + 1 ≤ 9 or x − 3 > 2

4x + 1 ≤ 9

4x ≤ 8

x ≤ 2

x − 3 > 2

x > 5

x ≤ 2 or x > 5

Answer: x ≤ 2 or x > 5

3
Solve and graph. −2x ≥ 6 or x + 5 > 8
Show Answer

−2x ≥ 6 or x + 5 > 8

−2x ≥ 6

x ≤ −3 (flip the sign)

x + 5 > 8

x > 3

x ≤ −3 or x > 3

Answer: x ≤ −3 or x > 3

4
Solve and graph. 3(x − 2) < 3 or 4x ≥ 20
Show Answer

3(x − 2) < 3 or 4x ≥ 20

3(x − 2) < 3

3x − 6 < 3

3x < 9

x < 3

4x ≥ 20

x ≥ 5

x < 3 or x ≥ 5

Answer: x < 3 or x ≥ 5

5
Solve and graph. 5 − x < 1 or 2x + 4 ≤ 0
Show Answer

5 − x < 1 or 2x + 4 ≤ 0

5 − x < 1

−x < −4

x > 4 (flip the sign)

2x + 4 ≤ 0

2x ≤ −4

x ≤ −2

x ≤ −2 or x > 4

Answer: x ≤ −2 or x > 4

Special Cases

Special Cases

AND and OR that collapse, plus no solution and all real numbers.

Example
Solve and graph. x > 2 and x > 6

x > 2 and x > 6

AND overlap → x > 6

Answer: x > 6

1
Solve and graph. x > 1 and x > 4
Show Answer

x > 1 and x > 4

AND overlap → x > 4

Answer: x > 4

2
Solve and graph. x > 1 or x > 4
Show Answer

x > 1 or x > 4

OR anything shaded works! → x > 1

Answer: x > 1

3
Solve and graph. x < 5 and x < 2
Show Answer

x < 5 and x < 2

AND overlap → x < 2

Answer: x < 2

4
Solve and graph. x < 5 or x < 2
Show Answer

x < 5 or x < 2

OR anything shaded works! → x < 5

Answer: x < 5

5
Solve and graph. x > 1 and x < −3
Show Answer

x > 1 and x < −3

AND means overlap. No overlap → no solution

Answer: no solution

6
Solve and graph. x > 1 or x < −3
Show Answer

x > 1 or x < −3

OR anything shaded works → infinitely many solutions

Answer: infinitely many solutions

7
Solve and graph. 2x + 9 > 2x + 4
Show Answer

2x + 9 > 2x + 4

9 > 4

Answer: all real numbers

8
Solve and graph. 6x − 5 < 6x − 12
Show Answer

6x − 5 < 6x − 12

−5 < −12

Answer: no solution

Super Challenge

Super Challenge

1
Solve and graph. 12x − 4 ≥ 14x + 1
Show Answer

12x − 4 ≥ 14x + 1

2x − 16 ≥ x + 4

x − 16 ≥ 4

x ≥ 20

Answer: x ≥ 20

2
Solve and graph. 2x + 3 > 9 and x + 5 > 10
Show Answer

2x + 3 > 9 and x + 5 > 10

2x + 3 > 9

2x > 6

x > 3

x + 5 > 10

x > 5

AND overlap → x > 5

Answer: x > 5

3
Solve and graph. 3x < 12 or x − 1 < 5
Show Answer

3x < 12 or x − 1 < 5

3x < 12

x < 4

x − 1 < 5

x < 6

OR anything shaded works! → x < 6

Answer: x < 6

4
Solve and graph. x + 4 > 6 and 2x < −8
Show Answer

x + 4 > 6 and 2x < −8

x + 4 > 6

x > 2

2x < −8

x < −4

AND means overlap. No overlap → no solution

Answer: no solution

5
Solve and graph. x + 4 > 6 or 2x < −8
Show Answer

x + 4 > 6 or 2x < −8

x + 4 > 6

x > 2

2x < −8

x < −4

OR anything shaded works → infinitely many solutions

Answer: infinitely many solutions

6
Solve and graph. 4x + 7 > 4x + 1
Show Answer

4x + 7 > 4x + 1

7 > 1

Answer: all real numbers

7
Solve and graph. 5x − 2 < 5x − 9
Show Answer

5x − 2 < 5x − 9

−2 < −9

Answer: no solution

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