Question

  

Solve and graph on a number line.

A. -3x < 30 +2x 

B. x + 3 ≥ 6(x -4) +7

Answer 1

  

B) The inequality is x + 3 ≥ 6(x - 4) + 7.

Apply Distributive Property of Multiplication over Subtraction: a(b - c) = ab - ac.

x + 3 ≥ 6(x) - 6(4) + 7

x + 3 ≥ 6x - 24 + 7

x + 3 ≥ 6x - 17

Apply Subtraction Property of Inequality: If a ≥ b then a - c ≥ b - c.

Subtract x from each side.

x + 3 - x ≥ 6x - 17 - x

Group like terms.

x - x + 3 ≥ 6x - x - 17

Combine like terms.

0 + 3 ≥ 5x - 17

3 ≥ 5x - 17

Apply Additon Property of Inequality: If a ≥ b then a + c ≥ b + c.

Add 17 to each side.

3 + 17 ≥ 5x - 17 + 17

20 ≥ 5x - 0

20 ≥ 5x

Apply Division Property of Inequality: If a ≥ b then a / c ≥ b / c.

Divide each side by 5.

20/5 ≥ 5x/5

Cancel common terms.

20/5 ≥ x

4 ≥ x

x ≤ 4.

Graph for the inequality x ≤ 4 is a follows:

Dot indicates that the number 4 is included.

  

Answer 2

A) The inequality is -3x < 30 + 2x.

Apply Subtraction Property of Inequality: If a < b then a - c < b - c.

Subtract 2x from each side.

-3x - 2x < 30 + 2x - 2x

-5x < 30 + 2x - 2x

Apply Additive Inverse Property: a + (- a) = 0.

-5x < 30 + 0

Apply Additive Identity Property: a + 0 = a.

-5x < 30

Apply Division Property of Inequality: If a < b then a / -c > b / -c.

Divide each side by -5.

(-5x) / (-5) > (30) / (-5)

Cancel common terms.

x > (30) / (-5)

x > -6.

Graph for the inequality x > -6 is a follows:

Circle indicates that the number -6 is not included.