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Inequalities: MI 2024 Promo Q2

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Millennia Institute2024 PromoPromo Q2●●○ Standard6 marks

What this question tests

Inequalities.

Question

Without using a calculator, solve the inequality $\frac{x^2 - 3x + 2}{x + 4} \geq 2.$ [4]
Hence solve the inequality $\frac{x^2 + 3x + 2}{4 - x} \geq 2.$ [2]

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(i) [4] $\begin{aligned} \frac{x^2-3x+2}{x+4}&\geq2\\ \frac{x^2-3x+2}{x+4}-2&\geq0\\ \frac{x^2-3x+2-2x-8}{x+4}&\geq0\\ \frac{x^2-5x-6}{x+4}&\geq0\\ \frac{(x+1)(x-6)}{x+4}&\geq0 \end{aligned}$

$Millennia Institute 2024 Promo Q2: solution diagram$

$-4<x\leq-1$ or $x\geq6$ .

2(ii) [4] $\frac{x^2+3x+2}{4-x}\geq2 \quad\Rightarrow\quad \frac{(-x)^2-3(-x)+2}{-x+4}\geq2$ Replace $x$ with $-x$ : $\begin{aligned} -4<-x\leq-1&\text{ or }-x\geq6\\ 4>x\geq1&\text{ or }x\leq-6\\ 1\leq x<4&\text{ or }x\leq-6 \end{aligned}$

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