(a) Solve the inequality : \(\frac{2}{5}(x – 2) – \frac{1}{6}(x + 5) \leq 0\).
(b) Given that P = \(\frac{x^{2} – y^{2}}{x^{2} + xy}\),
(i) express P in its simplest form ; (ii) find the value of P if x = -4 and y = -6.
Explanation
(a) \(\frac{2}{5}(x - 2) - \frac{1}{6}(x + 5) \leq 0\)
Multiplying through by the LCM of 5 and 6 (i.e 30)
\(12(x - 2) - 5(x + 5) \leq 0\)
\(12x - 24 - 5x - 25 \leq 0\)
\(7x - 49 \leq 0 \implies 7x \leq 49\)
\(x \leq 7\).
(b) (i) \(\frac{x^{2} - y^{2}}{x^{2} + xy}\)
P = \(\frac{(x - y)(x + y)}{x (x + y)}\)
P = \(\frac{x - y}{x}\)
(ii) When x = -4, y = -6
\(P = \frac{-4 - (-6)}{-4}\)
\(P = \frac{2}{-4}\)
\(P = - \frac{1}{2}\)