Solve |linear| = |linear|

Solve an equation where both sides are modulus of linear expressions, e.g. |3x+4| = |2x+5|.

22 questions · Moderate -0.5

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CAIE P2 2022 March Q1
3 marks Moderate -0.8
1 Solve the equation \(| 5 x - 2 | = | 4 x + 9 |\).
CAIE P2 2011 June Q1
3 marks Moderate -0.5
1 Solve the equation \(| 3 x + 4 | = | 2 x + 5 |\).
CAIE P2 2012 June Q1
4 marks Moderate -0.3
1 Solve the equation \(\left| x ^ { 3 } - 14 \right| = 13\), showing all your working.
CAIE P2 2013 June Q1
5 marks Moderate -0.3
1 Solve the equation \(\left| 2 ^ { x } - 7 \right| = 1\), giving answers correct to 2 decimal places where appropriate.
CAIE P3 2012 June Q1
3 marks Moderate -0.3
1 Solve the equation \(\left| 4 - 2 ^ { x } \right| = 10\), giving your answer correct to 3 significant figures.
CAIE P3 2013 June Q1
3 marks Moderate -0.8
1 Solve the equation \(| x - 2 | = \left| \frac { 1 } { 3 } x \right|\).
CAIE P2 2014 November Q1
3 marks Moderate -0.8
1 Solve the equation \(| 3 x - 1 | = | 2 x + 5 |\).
CAIE P2 2016 November Q1
3 marks Easy -1.2
1 Solve the equation \(| 0.4 x - 0.8 | = 2\).
Edexcel P3 2021 June Q6
8 marks Standard +0.3
6. Given that \(k\) is a positive constant,
  1. on separate diagrams, sketch the graph with equation
    1. \(y = k - 2 | x |\)
    2. \(y = \left| 2 x - \frac { k } { 3 } \right|\) Show on each sketch the coordinates, in terms of \(k\), of each point where the graph meets or cuts the axes.
  2. Hence find, in terms of \(k\), the values of \(x\) for which $$\left| 2 x - \frac { k } { 3 } \right| = k - 2 | x |$$ giving your answers in simplest form. \includegraphics[max width=\textwidth, alt={}, center]{76205772-5395-4ab2-96f9-ad9803b8388c-23_2647_1840_118_111}
OCR C3 2005 June Q2
4 marks Moderate -0.8
2 Find the exact solutions of the equation \(| 6 x - 1 | = | x - 1 |\).
OCR C3 2008 June Q1
4 marks Moderate -0.5
1 Find the exact solutions of the equation \(| 4 x - 5 | = | 3 x - 5 |\).
OCR MEI C3 2007 January Q1
5 marks Easy -1.2
1 Fig. 1 shows the graphs of \(y = | x |\) and \(y = | x - 2 | + 1\). The point P is the minimum point of \(y = | x - 2 | + 1\), and Q is the point of intersection of the two graphs. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{666dc19e-f293-4738-8530-fce90df23d17-2_490_844_493_607} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Write down the coordinates of P .
  2. Verify that the \(y\)-coordinate of Q is \(1 \frac { 1 } { 2 }\).
OCR MEI C3 Q1
4 marks Standard +0.3
1 Solve the equation \(| 3 - 2 x | = 4 | x |\).
OCR MEI C3 Q5
4 marks Moderate -0.5
5 Solve the equation \(| 2 x - 1 | = | x |\).
[0pt] [4]
OCR MEI C3 Q10
5 marks Moderate -0.8
10 Fig. 1 shows the graphs of \(y = | x |\) and \(y = | x - 2 | + 1\). The point P is the minimum point of \(y = | x - 2 | + 1\), and Q is the point of intersection of the two graphs. \begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{125b76c1-5ab3-4645-a3c4-cf167a04f453-3_491_833_503_657} \captionsetup{labelformat=empty} \caption{Fig. 1}
\end{figure}
  1. Write down the coordinates of P .
  2. Verify that the \(y\)-coordinate of Q is \(1 \frac { 1 } { 2 }\).
OCR MEI C3 2011 June Q1
4 marks Moderate -0.5
1 Solve the equation \(| 2 x - 1 | = | x |\).
OCR MEI C3 2014 June Q3
4 marks Moderate -0.3
3 Solve the equation \(| 3 - 2 x | = 4 | x |\).
OCR H240/03 2018 June Q2
3 marks Moderate -0.8
2 Solve the equation \(| 2 x - 1 | = | x + 3 |\).
AQA C3 2006 June Q4
8 marks Moderate -0.8
4
  1. Sketch and label on the same set of axes the graphs of:
    1. \(y = | x |\);
    2. \(y = | 2 x - 4 |\).
    1. Solve the equation \(| x | = | 2 x - 4 |\).
    2. Hence, or otherwise, solve the inequality \(| x | > | 2 x - 4 |\).
AQA C3 2013 June Q1
5 marks Moderate -0.3
1 The diagram below shows the graphs of \(y = | 2 x - 3 |\) and \(y = | x |\). \includegraphics[max width=\textwidth, alt={}, center]{063bbfa5-df49-44a1-8143-5e076397f63f-02_579_1150_351_482}
  1. Find the \(x\)-coordinates of the points of intersection of the graphs of \(y = | 2 x - 3 |\) and \(y = | x |\).
    (3 marks)
  2. Hence, or otherwise, solve the inequality $$| 2 x - 3 | \geqslant | x |$$ (2 marks)
OCR Pure 1 2018 March Q2
6 marks Moderate -0.8
2
  1. Given that \(| n | = 5\), find the greatest value of \(| 2 n - 3 |\), justifying your answer.
  2. Solve the equation \(| 3 x - 6 | = | x - 6 |\).
OCR Pure 1 2018 September Q3
6 marks Moderate -0.3
3
  1. The diagram below shows the graphs of \(y = | 3 x - 2 |\) and \(y = | 2 x + 1 |\). \includegraphics[max width=\textwidth, alt={}, center]{e3942549-bfc0-432a-bf49-7d01d44af01a-4_423_682_1110_694} On the diagram in your Printed Answer Booklet, give the coordinates of the points of intersection of the graphs with the coordinate axes.
  2. Solve the equation \(| 2 x + 1 | = | 3 x - 2 |\).