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2025 RI P2 Q1

2025 RI P2 Q1

8 marks
  1. Without using a calculator, solve the inequality \(\frac{x^2 - 5x + 6}{x^2 - 4} < \frac{2x - 3}{x + 2}\).[4]
    1. 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.[2]
    2. Hence solve the inequality \(\ln |x| < |x| - 5\).[2]

Solution:

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Answer:(a) \(x<-2\), \(0<x<2\) or \(x>2\). (b)(i) Vertical asymptote \(x=0\); intersections at \(x=\alpha\approx0.00678\), \(x=\beta\approx6.94\). (b)(ii) \(x<-\beta\), \(-\alpha<x<0\), \(0<x<\alpha\) or \(x>\beta\).

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