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Inequalities: DHS 2024 Promo Q7

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Dunman High School2024 PromoPromo P1 Q7●●● Challenging6 marks

What this question tests

Inequalities & Modulus.

Question

Given that both $a$ and $b$ are positive constants, on the same diagram, sketch the graphs of $y = \frac{2}{|x - a|} \qquad \text{and} \qquad y = b\,|x - a|.$ [3]
Hence, or otherwise, solve the inequality $\dfrac{2}{|x - a|} > b\,|x - a|$ . [3]

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(a)

$Dunman High School 2024 Promo Q7: solution diagram$

(b) $\begin{aligned} \frac2{|x-a|}&=b|x-a|\\ (x-a)^2&=\frac2b\\ x-a&=\pm\sqrt{\frac2b}\\ x&=a\pm\sqrt{\frac2b} \end{aligned}$ Hence, for $\dfrac2{|x-a|}>b|x-a|$ , $a-\sqrt{\frac2b}<x<a+\sqrt{\frac2b},\quad x\ne a$

Source marking note: With graph, to accept answer with symmetry deduction, to award full credit

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