✦ MA Intelligence →
Math Academy
You are here: Home / Questions / RI 2025 H2 Math Prelim Paper 2 — Inequalities (worked solution)

by

Inequalities: Rational & \(\ln|x|\) combined — RI 2025 H2 Math Prelim Paper 2

Raffles Institution2025 PrelimPaper 2●●● Challenging9 marks

What this question tests

Rational & \(\ln|x|\) combined.

Question

  • Without using a calculator, solve the inequality \[ \frac{x^2-5x+6}{x^2-4} \leq \frac{2x-3}{x+2}. \]

  • Sketch on the same diagram the graphs of \(y = \ln x\) and \(y = x - 5\), giving the equations of any asymptotes and the \(x\)-coordinates of the points of intersection between the two graphs.

  • Hence solve the inequality \(\ln|x| < |x| - 5\).

    • Show full worked solution▾

  • Solve \(\dfrac{x^2-5x+6}{x^2-4} \leq \dfrac{2x-3}{x+2}\).

    \[\begin{aligned} \frac{x^2-5x+6}{x^2-4} - \frac{2x-3}{x+2} &\leq 0\\[4pt] \frac{(x-2)(x-3)}{(x-2)(x+2)} - \frac{2x-3}{x+2} &\leq 0\\[4pt] \frac{(x-3)}{(x+2)} - \frac{2x-3}{x+2} &\leq 0 \quad (x \neq 2)\\[4pt] \frac{x-3-(2x-3)}{x+2} &\leq 0\\[4pt] \frac{-x}{x+2} &\leq 0\\[4pt] \frac{x}{x+2} &\geq 0 \end{aligned}\]

    So \(x < -2\) or \(x \geq 0\), excluding \(x = 2\).

  • Sketch of \(y = \ln x\) and \(y = x - 5\):

    Figure for this question

    Asymptote: \(x = 0\) (for \(y = \ln x\)).

    From GC, \(x\)-coordinates of intersection are \(x = 6.94\) (3 s.f.) or \(x = 0.00678\) (3 s.f.)

  • Figure for this question

    For \(\ln|x| < |x| - 5\),

    \[ x < -6.94 \text{ or } -0.00678 < x < 0 \text{ or } 0 < x < 0.00678 \text{ or } x > 6.94 \]

    OR

    \[ x < -6.94 \text{ or } -0.00678 < x < 0.00678 \text{ or } x > 6.94, \quad x \neq 0 \]
    • Answer: (a) \(x < -2 \quad\) or \(\quad 0 \leq x < 2 \quad\) or \(\quad x > 2\) (b)(ii) \(-0.00678 < x < 0,\quad 0 < x < 0.00678,\quad x < -6.94,\quad x > 6.94\)
    ← All Inequalities questionsMath Academy — JC H2 Maths tuition