(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...
Question 1 Report
(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.
Answer Details
(a) \(\dfrac25(x - 2) - \dfrac16(x + 5) \le 0\). Multiply through by the LCM \(30\):
\[12(x - 2) - 5(x + 5) \le 0\]
\[12x - 24 - 5x - 25 \le 0 \;\Rightarrow\; 7x - 49 \le 0 \;\Rightarrow\; 7x \le 49 \;\Rightarrow\; x \le 7\]
(b)(i) Factorise numerator and denominator of \(P = \dfrac{x^2 - y^2}{x^2 + xy}\):
\[P = \frac{(x - y)(x + y)}{x(x + y)} = \frac{x - y}{x}\]
(ii) With \(x = -4,\, y = -6\):
\[P = \frac{-4 - (-6)}{-4} = \frac{2}{-4} = -\frac12\]
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