O-Level Equations and inequalities: worked solution

Question

A rectangle has sides (3x−5) m and (x−10) m and area no more than 200 m². Find the range of x consistent with these conditions. [4]

The area of a rectangle is length × breadth, so Area = (3x - 5)(x - 10) "No more than 200 m²" means the area is at most 200, so (3x - 5)(x - 10) ≤ 200 Expand the left-hand side: 3x² - 30x - 5x + 50 ≤ 200 3x² - 35x + 50 ≤ 200 Bring everything to one side so the quadratic can be solved: 3x² - 35x - 150 ≤ 0 First find where the left-hand side equals 0, using the formula with a = 3, b = -35, c = -150: discriminant = (-35)² - 4(3)(-150) = 1225 + 1800 = 3025, and √3025 = 55 x = (35 ± 55)/6 x = 90/6 = 15 or x = -20/6 = -10/3 The graph of 3x² - 35x - 150 is a U-shape (the x² coefficient 3 is positive), so the expression is ≤ 0 between the two roots: -10/3 ≤ x ≤ 15 This is not the whole answer, because the two sides must be real lengths, so each must be positive: 3x - 5 > 0 gives x > 5/3 x - 10 > 0 gives x > 10 The stricter of the two is x > 10, so that is the condition to use. Combine x > 10 with -10/3 ≤ x ≤ 15. The overlap of the two is 10 < x ≤ 15