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Inequalities: VJC 2024 Promo Q2

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Victoria Junior College2024 PromoPromo Q2●○○ Foundational5 marks

What this question tests

Solving inequalities.

(a)Without using a calculator, solve the inequality $\dfrac{2x^2+3x+10}{x^2-x}\geq2$ . [3]

(b)Hence, find the set of values of $x$ for which $\dfrac{2e^{2x}-3e^x+10}{e^{2x}+e^x}\geq2$ . [2]

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(a)

$\begin{aligned} \frac{2x^2+3x+10}{x^2-x}&\geq2\\ \frac{2x^2+3x+10-2(x^2-x)}{x^2-x}&\geq0\\ \frac{5x+10}{x(x-1)}&\geq0 \end{aligned}$

$Victoria Junior College 2024 Promo Q2: solution diagram$

$\therefore -2\leq x<0\quad\text{or}\quad x>1$

(b) Replace $x$ in part (a) with $-e^x$

$-2\leq-e^x<0\quad\text{or}\quad-e^x>1$ (Reject since $-e^x<0$ for all real $x$ )

$0<e^x\leq2$

$x\leq\ln2\qquad\therefore\{x\in\mathbb R:x\leq\ln2\}$