SOLVING LINEAR INEQUALITIES IN ONE VARIABLE

Solve each inequality.

Problem 1 :

4x - 1 ≥ 7

Solution :

4x - 1 ≥ 7

Add 1 to both sides.

4x – 1 + 1 ≥ 7 + 1

4x ≥ 8

Divide both sides by 4.

4x/4 ≥ 8/4

x ≥ 2

Problem 2 :

2(x – 5) ≤ 8

Solution :

2(x – 5) ≤ 8

Use distributive property.

2x – 10 ≤ 8

Add 10 to both sides.

2x – 10 + 10 ≤ 8 + 10

2x ≤ 18

Divide both sides by 2.

2x/2 ≤ 18/2

x ≤ 9

Problem 3 :

3 – 2x < x + 6

Solution :

3 – 2x < x + 6

Subtract x from both sides.

3 – 2x – x < x - x + 6

3 – 3x < 6

Subtract 3 from both sides.

3 – 3 - 3x < 6 – 3

-3x < 3

Divide both sides by 3.

-3/3x < 3/3

-x < 1

x > -1

Problem 4 :

(1/2)x > 5

Solution :

(1/2)x > 5

Multiply 2 on both sides.

(1/2)x × 2 > 5 × 2

x > 10 

Problem 5 :

3(x + 4) > 12

Solution :

3(x + 4) > 12

Use distributive property.

3x + 12 > 12

Subtract 12 from both sides.

3x + 12 - 12 > 12 – 12

3x > 0

Divide both sides by 3.

3x/3 > 0/3

x > 0

Problem 6 :

2x – 7 ≤ 5 – 4x

Solution :

2x – 7 ≤ 5 – 4x

Add 4x to both sides.

2x – 7 + 4x ≤ 5 – 4x + 4x

6x – 7 ≤ 5

Add 7 to both sides.

6x – 7 + 7 ≤ 5 + 7

6x ≤ 12

Divide both sides by 6.

6x/6 ≤ 12/6

x ≤ 2

Problem 7 :

3x + 2 < 11

Solution :

3x + 2 < 11

Subtract 2 from both sides.

3x + 2 - 2 < 11 – 2

3x < 9

Divide both sides by 3.

3x/3 < 9/3

x < 3

Problem 8 :

4(x – 6) ≥ 20

Solution :

4(x – 6) ≥ 20

Use distributive property.

4x – 24 ≥ 20

Add 24 to both sides.

4x – 24 + 24 ≥ 20 + 24

4x ≥ 44

Divide both sides by 4.

4x/4 ≥ 44/4

x ≥ 11

Problem 9 :

(1/4(x < 2

Solution :

(1/4)x < 2

Multiply 4 on both sides.

(1/4)x × 4 < 2 × 4

x < 8

Problem 10 :

12 – 3x > 2x + 1

Solution :

12 – 3x > 2x + 1

Subtract 2x from both sides.

12 – 3x – 2x > 2x – 2x + 1

12 – 5x > 1

Add 5x to both sides.

12 – 5x + 5x > 1 + 5x

12 > 1 + 5x

Subtract 1 from both sides.

12 - 1 > 1 - 1 + 5x

11 > 5x

Divide both sides by 5.

11/5 > 5/5x

11/5 > x

x < 11/5

Problem 11 :

(x – 5)/7 ≤ -3

Solution :

(x – 5)/7 ≤ -3

Multiply 7on both sides.

(x – 5)/7 × 7 ≤ -3 × 7

 x – 5 ≤ -21

Add 5 to both sides.

x – 5 + 5 ≤ -21 + 5

x ≤ -16

Problem 12 :

3(5 – x) ≥ 7x - 1

Solution :

3(5 – x) ≥ 7x - 1

Use distributive property.

15 – 3x ≥ 7x – 1

Add 3x to both sides.

15 - 3x + 3x ≥ 7x – 1 + 3x

15 ≥ 10x - 1

Add 1to both sides.

15 + 1 ≥ 10x – 1 + 1

16 ≥ 10x

Divide both sides by 10.

16/10 ≥ 10x/10

x ≤ 1.6

Problem 13 :

3y – (2y + 2) ≤ 7

Solution :

3y – (2y + 2) ≤ 7

3y – 2y - 2 ≤ 7

y – 2 ≤ 7

Add 2 to both sides.

y – 2 + 2 ≤ 7 + 2

y ≤ 9

Problem 14 :

(m + 2)/5 < 2m/3

Solution :

(m + 2)/5 < 2m/3

(m + 2)/5 - 2m/3 < 0

(3(m + 2))/(5 × 3) – ((2m)5)/(3 × 5) < 0

(3m + 6)/15 – 10m/15 < 0

3m + 6 – 10m < 0

3m + 6 < 10m

Subtract 3m from both sides.

3m – 3m + 6 < 10m – 3m

6 < 7m

Divide both sides by 7.

6/7 < 7/7m

6/7 < m

m > 6/7

Problem 15 :

(m – 2)/3 ≥ (2m + 1)/7

Solution :

(m – 2)/3 ≥ (2m + 1)/7

(m – 2)/3 + (2m + 1)/7 ≥ 0

(7(m – 2))/(3 × 7) + (3(2m + 1))/(7 × 3) ≥ 0

(7m – 14)/21 + (6m + 3)/21 ≥ 0

7m – 14 + 6m + 3 ≥ 0

7m – 14 ≥ 6m + 3

Add 14 to both sides.

7m – 14 + 14 ≥ 6m + 3 + 14

7m ≥ 6m + 17

Subtract 6m from both sides.

7m – 6m ≥ 6m - 6m + 17

m ≥ 17