Solve |f(x)| compared to |g(x)| with parameters: equation or inequality only

Solve a modulus equation or inequality where expressions contain a positive constant parameter a or k, giving answer in terms of that parameter, without requiring a sketch.

10 questions · Standard +0.6

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CAIE P2 2017 June Q1
3 marks Standard +0.3
1 Solve the equation \(| x + a | = | 2 x - 5 a |\), giving \(x\) in terms of the positive constant \(a\).
CAIE P3 2010 June Q1
4 marks Challenging +1.2
1 Solve the inequality \(| x + 3 a | > 2 | x - 2 a |\), where \(a\) is a positive constant.
CAIE P3 2018 November Q1
4 marks Standard +0.8
1 Find the set of values of \(x\) satisfying the inequality \(2 | 2 x - a | < | x + 3 a |\), where \(a\) is a positive constant. [4]
CAIE P3 2022 June Q1
4 marks Challenging +1.2
1 Find, in terms of \(a\), the set of values of \(x\) satisfying the inequality $$2 | 3 x + a | < | 2 x + 3 a |$$ where \(a\) is a positive constant.
CAIE P3 2021 November Q2
4 marks Challenging +1.2
2 Solve the inequality \(| 3 x - a | > 2 | x + 2 a |\), where \(a\) is a positive constant.
OCR C3 2011 January Q1
3 marks Moderate -0.8
1 Solve the equation \(| 3 x + 4 a | = 5 a\), where \(a\) is a positive constant.
CAIE P3 2014 June Q1
4 marks Standard +0.8
Find the set of values of \(x\) satisfying the inequality $$|x + 2a| > 3|x - a|,$$ where \(a\) is a positive constant. [4]
CAIE P3 2018 November Q1
4 marks Standard +0.8
Find the set of values of \(x\) satisfying the inequality \(2|2x - a| < |x + 3a|\), where \(a\) is a positive constant. [4]
OCR H240/03 2021 November Q4
5 marks Moderate -0.3
  1. Sketch, on a single diagram, the following graphs.
    [2]
  2. Hence explain why the equation \(x|x - 1| = k\) has exactly one real root for any negative value of \(k\). [1]
  3. Determine the real root of the equation \(x|x - 1| = -6\). [2]
SPS SPS FM Pure 2024 January Q2
6 marks Standard +0.3
  1. Find, in terms of \(k\), the set of values of \(x\) for which $$k - |2x - 3k| > x - k$$ giving your answer in set notation. [4]
  2. Find, in terms of \(k\), the coordinates of the minimum point of the graph with equation $$y = 3 - 5f\left(\frac{1}{2}x\right)$$ where $$f(x) = k - |2x - 3k|$$ [2]