1.02l Modulus function: notation, relations, equations and inequalities

1 Solve the inequality \(| x + 1 | < | 3 x + 5 |\).

1 Use the trapezium rule with four intervals to find an approximation to $$\int _ { 1 } ^ { 5 } \left| 2 ^ { x } - 8 \right| \mathrm { d } x$$

1 Solve the equation \(| 3 x - 1 | = | 2 x + 5 |\).

1

1. Solve the equation \(| 3 x - 2 | = 5\).
2. Hence, using logarithms, solve the equation \(\left| 3 \times 5 ^ { y } - 2 \right| = 5\), giving the answer correct to 3 significant figures.

2

1. Solve the equation \(| 2 x + 3 | = | x + 8 |\).
2. Hence, using logarithms, solve the equation \(\left| 2 ^ { y + 1 } + 3 \right| = \left| 2 ^ { y } + 8 \right|\). Give the answer correct to 3 significant figures.

1 Solve the equation \(| 0.4 x - 0.8 | = 2\).

3 It is given that the variable \(x\) is such that $$1.3 ^ { 2 x } < 80 \quad \text { and } \quad | 3 x - 1 | > | 3 x - 10 | .$$ Find the set of possible values of \(x\), giving your answer in the form \(a < x < b\) where the constants \(a\) and \(b\) are correct to 3 significant figures.

2 It is given that \(x\) satisfies the equation \(| x + 1 | = 4\). Find the possible values of $$| x + 4 | - | x - 4 | .$$

1

1. Solve the equation \(| 9 x - 2 | = | 3 x + 2 |\).
2. Hence, using logarithms, solve the equation \(\left| 3 ^ { y + 2 } - 2 \right| = \left| 3 ^ { y + 1 } + 2 \right|\), giving your answer correct to 3 significant figures.

1 Solve the inequality \(| 3 x - 5 | < 2 | x |\).

1

1. Solve the inequality \(| 2 x - 7 | < | 2 x - 9 |\).
2. Hence find the largest integer \(n\) satisfying the inequality \(| 2 \ln n - 7 | < | 2 \ln n - 9 |\).

2

1. Solve the equation \(| 4 x + 5 | = | x - 7 |\).
2. Hence, using logarithms, solve the equation \(\left| 2 ^ { y + 2 } + 5 \right| = \left| 2 ^ { y } - 7 \right|\), giving the answer correct to 3 significant figures.

1 Candidates answer on the Question Paper.
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The total number of marks for this paper is 50. 1

1. Solve the inequality \(| 2 x - 7 | < | 2 x - 9 |\).
2. Hence find the largest integer \(n\) satisfying the inequality \(| 2 \ln n - 7 | < | 2 \ln n - 9 |\).

1 Solve the inequality \(| 2 x - 1 | > 3 | x + 2 |\).

1 Solve the inequality \(2 | 3 x - 1 | < | x + 1 |\).

1 Solve the inequality \(| 2 x - 1 | < 3 | x + 1 |\).

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.

2

1. Sketch the graph of \(y = | 2 x + 3 |\).
2. Solve the inequality \(3 x + 8 > | 2 x + 3 |\).

1 Solve the inequality \(| 5 x - 3 | < 2 | 3 x - 7 |\).

1

1. Sketch the graph of \(\mathrm { y } = | \mathrm { x } - 2 \mathrm { a } |\), where \(a\) is a positive constant.
2. Solve the inequality \(2 \mathrm { x } - 3 \mathrm { a } < | \mathrm { x } - 2 \mathrm { a } |\).

2 Solve the equation \(\ln 3 + \ln ( 2 x + 5 ) = 2 \ln ( x + 2 )\). Give your answer in a simplified exact form.

1 Solve the inequality \(| 2 x + 3 | > 3 | x + 2 |\).

1 Solve the inequality \(2 - 5 x > 2 | x - 3 |\).

1 Solve the equation \(4 \left| 5 ^ { x } - 1 \right| = 5 ^ { x }\), giving your answers correct to 3 decimal places.

2 Solve the inequality \(| 3 x - a | > 2 | x + 2 a |\), where \(a\) is a positive constant.