(a) Solve the inequality : (4 + frac{3}{4}(x + 2) leq frac{3}{8}x + 1)
(b)
The diagram shows a rectangle PQRS from which a square of side x cm has been cut. If the area of the shaded portion is 484(cm^{2}), find the values of x.
Explanation
(a) (4 + frac{3}{4}(x + 2) leq frac{3}{8}x + 1)
Multiply through by the LCM of 4 and 8, to clear fractions :
(8 times 4 + 8 times frac{3}{4}(x + 2) leq 8 times frac{3}{8}x + 8 )
(32 + 6(x + 2) leq 3x + 8 )
(32 + 6x + 12 leq 3x + 8 )
(44 + 6x leq 3x + 8)
(6x – 3x leq 8 – 44 implies 3x leq -36)
(x leq – 12).
(b)
Area ((A_{1})), of rectangle PQRS = (20(20 + x) cm^{2})
Area ((A_{2})), of square = (x times x = x^{2} cm^{2})
(implies 20(20 + x) – x^{2} = 484)
(400 + 20x – x^{2} = 484)
(x^{2} – 20x + 484 – 400 = x^{2} – 20x + 84 = 0)
(x^{2} – 14x – 6x + 84 = 0)
(x( x – 14) – 6(x – 14) = 0 )
((x – 14)(x – 6) = 0)
(therefore text{x = 6 cm or 14 cm})