Inequalities: MI 2024 Promo Q2

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

$-4<x\leq-1$ 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 $-x$: $$\begin{aligned} -4<-x\leq-1&\text{ or }-x\geq6\\ 4>x\geq1&\text{ or }x\leq-6\\ 1\leq x<4&\text{ or }x\leq-6 \end{aligned}$$