Solve and graph the inequality

Question

1. 4x-5<_ 15

2. 3<2x+11<17

3. |3x-1|>7

4. |1-2x|<_ 3?

Answer 1

1) The inequality 4x - 5 ≤ 15

Add 5 to each side of inequality.

4x - 5 + 5 ≤ 15 + 5

4x ≤ 20

Divide each side of inequality by 4.

4x/4 ≤ 20/4

x ≤ 5

Solution set is { x | x ≤ 5 }

Graph 

Note that closed circle means 5 is including in the solution set.

Answer 2

2) The inequality 3 < 2x + 11 < 17

3 - 11 < 2x + 11 - 11 < 17 - 11

- 8 < 2x < 6

( -8)/2 < 2x/2 < 6/2

- 4 < x < 3

Solution set is { x| - 4 < x < 3 }

Graph

Solution set on a number line.

Note that open circle means -4 and 3 are not included in the solution set.

Answer 3

3) The absolute inequality | 3x - 1| > 7

|x| > a is equivalent to x > a or x < - a

Case 1 :

3x - 1 > 7

3x > 7 + 1

3x > 8

x > 8/3

Case 2 :

3x - 1 < - 7

3x < - 7 + 1

3x < - 6

x < (-6/3)

x < - 2

Solution set { x | x < - 2 or x > 8/3 }

Graph the solution set for each inequality and find their intersection.

Answer 4

4) The absolute inequality | 1 - 2x | ≤ 3

| x | ≤ a is equivalent to - a ≤ x ≤ a

- 3 ≤ 1 - 2x ≤ 3

- 3 - 1 ≤ - 1 + 1 - 2x ≤ 3 - 1

- 4 ≤ - 2x ≤ 2

Divide each side of inequality by negative 2 and flip the symbols.

( - 4)/(-2) ≥ ( -2x)/(-2) ≥ 2/(-2)

2 ≥ x ≥ - 1

Solution set is { x | - 1 ≤ x ≤ 2 }

Graph the solution set for each inequality and find their intersection.

Note that closed circle means -1 and 2 are inclede in the solution set.