Chapter 2: The modulus function
Miscellaneous exercise 2 1 Solve the inequality |3x − 1| < 2. (Cambridge AS & A Level Mathematics 9709/02 Paper 2 Q1 June 2008) 2 Solve the inequality |x − 3| > |2x|. (Cambridge AS & A Level Mathematics 9709/02 Paper 2 Q1 November 2008) 3 Solve the equation |3x + 4| = |2x + 5|. (Cambridge AS & A Level Mathematics 9709/21 Paper 2 Q1 June 2011) 4 Solve the inequality |4 – 5x| < 3. (Cambridge AS & A Level Mathematics 9709/21 Paper 2 Q1 November 2011) 5 Solve the equation |x 3 – 14| = 13, showing all your working. (Cambridge AS & A Level Mathematics 9709/21 Paper 2 Q1 June 2012) 6 Solve the equation 4|x| = |x − 1|. On the same diagram sketch the graphs of y = 4|x| and y = |x − 1|, and hence, or otherwise, solve the inequality 4|x| > |x − 1|. 7 Solve the inequality |x – 2| ≥ |x + 5|. (Cambridge AS & A Level Mathematics 9709/21 Paper 2 Q1 November 2012) 8 Solve the inequality |x + 1| < |3x + 5|. (Cambridge AS & A Level Mathematics 9709/21 Paper 2 Q1 November 2013) 9 Solve the inequality |x| < 4|x − 3|. 10 Rewrite the function k(x) defined by k(x) = |x + 3| + |4 − x| for the following three cases, without using the modulus in your answer. a x>4
b −3 ≤ x ≤ 4
c x < −3
11 Solve the equations a x + 2x − 1 = 3, 12 i
b 3 + |2x – 1| = x.
Solve the inequality |y – 5| < 1.
ii Hence solve the inequality |3x – 5| < 1, giving 3 significant figures in your answer. (Cambridge AS & A Level Mathematics 9709/02 Paper 2 Q3 November 2007) 13 Solve the inequality |x – 2| > 3|2x + 1|. (Cambridge AS & A Level Mathematics 9709/03 Paper 3 Q1 June 2008) 14 Sketch the graphs of the following functions. a y=
x°
b y=
x°
c y=
x°
15 Solve the inequality 2 – 3x < |x – 3|. (Cambridge AS & A Level Mathematics 9709/31 Paper 3 Q1 November 2009)
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