Solve f(x) > g(x) using sketch

Questions requiring sketching rational function(s) and solving inequalities by comparing f(x) with another function g(x) (linear or otherwise), typically finding intersection points.

5 questions · Moderate -0.3

1.02n Sketch curves: simple equations including polynomials
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AQA Further AS Paper 1 2024 June Q10
6 marks Standard +0.3
10
  1. Write down the equations of the asymptotes of \(C\) 10
  2. The line \(L\) has equation $$y = - \frac { 2 } { 5 } x + 2$$ 10 (b) (i) Draw the line \(L\) on Figure 1 10 (b) (ii) Hence, or otherwise, solve the inequality $$\frac { 2 x - 10 } { 3 x - 5 } \leq - \frac { 2 } { 5 } x + 2$$
AQA FP1 2014 June Q6
10 marks Standard +0.3
A curve \(C\) has equation \(y = \frac{1}{x(x + 2)}\).
  1. Write down the equations of all the asymptotes of \(C\). [2 marks]
  2. The curve \(C\) has exactly one stationary point. The \(x\)-coordinate of the stationary point is \(-1\).
    1. Find the \(y\)-coordinate of the stationary point. [1 mark]
    2. Sketch the curve \(C\). [2 marks]
  3. Solve the inequality $$\frac{1}{x(x + 2)} \leqslant \frac{1}{8}$$ [5 marks]
AQA FP1 2016 June Q9
11 marks Standard +0.3
A curve \(C\) has equation \(y = \frac{x - 1}{(x - 2)(2x - 1)}\). The line \(L\) has equation \(y = \frac{1}{2}(x - 1)\).
  1. Write down the equations of the asymptotes of \(C\). [2 marks]
  2. By forming and solving a suitable cubic equation, find the \(x\)-coordinates of the points of intersection of \(L\) and \(C\). [3 marks]
  3. Given that \(C\) has no stationary points, sketch \(C\) and \(L\) on the same axes. [3 marks]
  4. Hence solve the inequality \(\frac{x - 1}{(x - 2)(2x - 1)} \geqslant \frac{1}{2}(x - 1)\). [3 marks]
SPS SPS FM 2024 October Q2
5 marks Easy -1.2
  1. Sketch the curve with equation $$y = \frac{k}{x} \quad x \neq 0$$ where \(k\) is a positive constant. [2]
  2. Hence or otherwise, solve $$\frac{16}{x} \leq 2$$ [3]
SPS SPS SM 2023 October Q7
5 marks Easy -1.3
  1. Sketch the curve with equation $$y = \frac{k}{x} \quad x \neq 0$$ where \(k\) is a positive constant. [2]
  2. Hence or otherwise, solve $$\frac{16}{x} \leq 2$$ [3]