Question

Solve and graph on a number line. A. -3x < 30 +2x B. x + 3 ≥ 6(x -4) +7 B. Find the length and the width of a rectangle when the width is 4 ft shorter than the length. The perimeter of the rectangle is greater than 72 ft.

Answer 1

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

Subtract 2x from each side.

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

- 5x < 30

Divide each side by negative 5 and reverse the inequality symbol.

(- 5x) / (- 5) > (5*6) / (- 5)

Cancel common terms.

x > - 6.

The solution of the inequality is x > - 6 and its graph is

The open circle means that - 6 is not a solution of the inequality.

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

Apply distributive property : a(b - c) = ab - ac.

x + 3 ≥ 6x - 24 +7

x + 3 ≥ 6x - 17

Subtract 3 from each side.

x + 3 - 3 ≥ 6x - 17 - 3

x ≥ 6x - 20

Subtract 6x from each side.

x - 6x ≥ 6x - 20 - 6x

- 5x ≥ - 20

Divide each side by negative 5 and reverse the inequality symbol.

(- 5x) / (- 5) ≤ (- 5*4) / (- 5)

Cancel common terms.

x ≤ 4.

The solution of the inequality is x ≤ 4 and its graph is

The closed circle means that 4 is a solution of the inequality.

• The formula for the perimeter of the rectangle P = 2(l + w), where l = length and w = width.

Let x be the length(l) of the rectangle.

The width(w) of the rectangle = 4 ft shorter than the length = x - 4.

The perimeter(P) of the rectangle is greater than 72 ft ⇒ P > 72.

2(l + w) > 72

2[ x + (x - 4) ] > 72

x + (x - 4) > 36

2x - 4 > 36

2x > 40

x > 20.

The length(l) of the rectangle is greater than 20 ft ⇒ l = x > 20.

Subtract 4 from each side of the above inequality.

x - 4 > 20 - 4

x - 4 > 16.

The width(w) of the rectangle is greater than 16 ft ⇒ w = x - 4 > 16.

Length : l  > 20 and Width : w  > 16.