# Solve Inequalities (1)

In this worksheet, students must solve simple two-stage inequalities.

Key stage:  KS 3

Curriculum topic:   Algebra

Curriculum subtopic:   Understand Expressions, Equations, Inequalities, Terms and Factors

Difficulty level:

### QUESTION 1 of 10

When we solve algebraic inequalities, our aim is to end up with one letter on one side of the inequality sign and one number on the other.  This is the solution.

Just as in solving equations, we do this by using inverse operations to undo things that get in the way, but remember that we must do the same thing to both sides.

If you reverse an inequality, remember to reverse the sign.

5 < 8

8 > 5

Example

5b + 3 > 23

Subtract 3 from both sides.

5b + 3 - 3 > 23 - 3

Simplify

5b > 20

Divide both sides by 5

5b ÷ 5 > 20 ÷ 5

Simplify

b > 4

Example

b/5 - 3 ≤ 17

Add 3 to both sides.

b/5 - 3 + 3 ≤ 17 + 3

Simplify

b/5 ≤ 20

Multiply both sides by 5

b ÷ 5 x 5 ≤ 20 x 5

Simplify

b ≤ 100

Example

Solve for a

5 - a ≥ 10

Add a to both sides.

5 - a + a ≥ 10 + a

Simplify

5 ≥ 10 + a

Subtract 10 from both sides.

5 -10 ≥ 10 - 10 + a

Simplify

-5 ≥ a

REVERSE and BE CAREFUL to change the SIGN DIRECTION

a -5

Solve for a:

3a + 4 < 19

a < 23/3

a < 23

a < 5

Solve for a:

3a - 4 < 5

a < ⅓

a < 27

a < 3

Solve for a:

7a + 2 ≥ 30

a ≥ 23

a ≥ 4

a = 4

Solve for a:

4a - 2 ≥ 22

a ≥ 24

a ≥ 5

a ≥ 6

Solve for a:

a/2 + 3 < 7

a < 8

a < 2

a < 20

Solve for a:

a/3 - 7 > 2

a > 27

a > 3

a > 15

Solve for a:

5a - 8 ≤ 92

a ≤ 95

a ≤ 20

a ≤ 84

Solve for a:

10 - a < 20

a < 10

a < -10

a > -10

Solve for a:

15 - 5a > 20

a < -1

a > -1

a > 1

Solve for a:

13 - 7a ≤ 20

a ≤ ½

a ≥ -1

a ≥ 1

• Question 1

Solve for a:

3a + 4 < 19

a < 5
EDDIE SAYS

3a + 4 < 19

Subtract 4 from both sides

3a + 4 - 4 < 19 - 4

3a  < 15

Divide both sides by 3

3a  < 15

3a ÷ 3 < 15 ÷ 3

a < 5

• Question 2

Solve for a:

3a - 4 < 5

a < 3
EDDIE SAYS

3a - 4 < 5

Add 4 to both sides

3a - 4 + 4 < 5 + 4

3a  < 9

Divide both sides by 3

3a  < 9

3a ÷ 3 < 9 ÷ 3

a < 3

• Question 3

Solve for a:

7a + 2 ≥ 30

a ≥ 4
EDDIE SAYS

7a + 2 ≥ 30

Subtract 2 from both sides

7a + 2 - 2 ≥ 30 - 2

7a ≥ 28

Divide both sides by 7

7a ≥ 28

7a ÷ 7 ≥ 28 ÷ 7

a ≥ 4

• Question 4

Solve for a:

4a - 2 ≥ 22

a ≥ 6
EDDIE SAYS

4a - 2 ≥ 22

Add two to both sides

4a - 2 + 2 ≥ 22 + 2

4a ≥ 24

Divide both sides by 4

4a ≥ 24

4a ÷ 4 ≥ 22 ÷ 4

a ≥ 6

• Question 5

Solve for a:

a/2 + 3 < 7

a < 8
EDDIE SAYS

a/2 + 3 < 7

Subtract 3 from both sides

a/2 +3 - 3 < 7 - 3

a/2 < 4

Multiply both sides by 2

a/2 < 4

a/2 x 2 < 4 x 2

a < 8

• Question 6

Solve for a:

a/3 - 7 > 2

a > 27
EDDIE SAYS

a/3 - 7 > 2

Add 7 to both sides

a/3 -7 + 7 > 2 + 7

a/3 > 9

Multiply both sides by 3

a/3 > 9

a/3 x 3 > 9 x 3

a > 27

• Question 7

Solve for a:

5a - 8 ≤ 92

a ≤ 20
EDDIE SAYS

5a - 8 ≤ 92

Add 8 to both sides

5a - 8 + 8 ≤ 92 + 8

5a ≤ 100

Divide both sides by 5

5a ≤ 100

5a ÷ 5 ≤ 100 ÷ 5

a ≤ 20

• Question 8

Solve for a:

10 - a < 20

a > -10
EDDIE SAYS

10 - a < 20

Subtract 10 from both sides

10 - a -10 < 20 -10

- a < 10

Since we want a, not -a, we need to change the signs.

Don't forget that if you do this, you have to switch the sign.

a > -10

• Question 9

Solve for a:

15 - 5a > 20

a < -1
EDDIE SAYS

15 - 5a > 20

Subtract 15 from both sides

15 - 5a -15 > 20 -15

- 5a > 5

Divide both sides by -5

Don't forget that this will change the inequality

-5a ÷ -5 > 5 ÷ -5

a < -1

• Question 10

Solve for a:

13 - 7a ≤ 20

a ≥ -1
EDDIE SAYS

13 - 7a ≤ 20

Subtract 13 from both sides

13 - 7a -13 ≤ 20 - 13

- 7a ≤ 7

Divide both sides by - 7

Don't forget that this will change the inequality

- 7a ÷ -7 ≤ 7 ÷ -7

a ≥ -1

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