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

4

1. Sketch, on the same diagram, the graphs of \(y = | 3 x + 2 a |\) and \(y = | 3 x - 4 a |\), where \(a\) is a positive constant. Give the coordinates of the points where each graph meets the axes.
2. Find the coordinates of the point of intersection of the two graphs.
3. Deduce the solution of the inequality \(| 3 x + 2 a | < | 3 x - 4 a |\).

5

1. Sketch, on the same diagram, the graphs of \(y = | 2 x - 3 |\) and \(y = 3 x + 5\).
2. Solve the inequality \(3 x + 5 < | 2 x - 3 |\).

1 Solve the inequality \(| 3 x - 7 | < | 4 x + 5 |\).

2 The solutions of the equation \(5 | x | = 5 - 2 x\) are \(x = a\) and \(x = b\), where \(a < b\).
Find the value of \(| 3 a - 1 | + | 7 b - 1 |\).

5

1. By sketching the graphs of $$y = | 5 - 2 x | \quad \text { and } \quad y = 3 \ln x$$ on the same diagram, show that the equation \(| 5 - 2 x | = 3 \ln x\) has exactly two roots.
2. Show that the value of the larger root satisfies the equation \(x = 2.5 + 1.5 \ln x\).
3. Show by calculation that the value of the larger root lies between 4.5 and 5.0.
4. Use an iterative formula, based on the equation in part (b), to find the value of the larger root correct to 3 significant figures. Give the result of each iteration to 5 significant figures.

4 The polynomial \(\mathrm { p } ( x )\) is defined by $$\mathrm { p } ( x ) = 2 x ^ { 3 } + 3 x ^ { 2 } + k x - 30$$ where \(k\) is a constant. It is given that \(( x - 3 )\) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(k\).
2. Hence find the quotient when \(\mathrm { p } ( x )\) is divided by ( \(x - 3\) ) and factorise \(\mathrm { p } ( x )\) completely.
3. It is given that \(a\) is one of the roots of the equation \(\mathrm { p } ( x ) = 0\). Given also that the equation \(| 4 y - 5 | = a\) is satisfied by two real values of \(y\), find these two values of \(y\).

3

1. Sketch on the same diagram the graphs of \(y = | 3 x - 8 |\) and \(y = 5 - x\).
2. Solve the inequality \(| 3 x - 8 | < 5 - x\).
3. Hence determine the largest integer \(N\) satisfying the inequality \(\left| 3 e ^ { 0.1 N } - 8 \right| < 5 - e ^ { 0.1 N }\).

1 Solve the inequality \(| 5 x + 7 | > | 2 x - 3 |\). \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-02_67_1653_333_244} \includegraphics[max width=\textwidth, alt={}, center]{76df3465-9617-4f2b-a8b7-f474b2817504-02_2715_37_143_2010}

5

1. Sketch, on the same diagram, the graphs of \(y = | x + 2 k |\) and \(y = | 2 x - 3 k |\), where \(k\) is a positive constant. Give, in terms of \(k\), the coordinates of the points where each graph meets the axes.
2. Find, in terms of \(k\), the coordinates of each of the two points where the graphs intersect.
3. Find, in terms of \(k\), the largest value of \(t\) satisfying the inequality $$\left| 2 ^ { t } + 2 k \right| \geqslant \left| 2 ^ { t + 1 } - 3 k \right| .$$

1

1. Sketch, on the same diagram, the graphs of \(y = | 3 x - 5 |\) and \(y = x + 2\).
2. Solve the equation \(| 3 x - 5 | = x + 2\).

1 Solve the equation \(| 5 x - 2 | = | 4 x + 9 |\).

4

1. Solve the equation \(| 2 x - 5 | = | x + 6 |\).
2. Hence find the value of \(y\) such that \(\left| 2 ^ { 1 - y } - 5 \right| = \left| 2 ^ { - y } + 6 \right|\). Give your answer correct to 3 significant figures.

2

1. Sketch, on the same diagram, the graphs of \(y = x + 3\) and \(y = | 2 x - 1 |\).
2. Solve the equation \(x + 3 = | 2 x - 1 |\).
3. Find the value of \(y\) such that \(5 ^ { \frac { 1 } { 2 } y } + 3 = \left| 2 \times 5 ^ { \frac { 1 } { 2 } y } - 1 \right|\). Give your answer correct to 3 significant figures.

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

2 The solutions of the equation \(| 4 x - 1 | = | x + 3 |\) are \(x = p\) and \(x = q\), where \(p < q\).
Find the exact values of \(p\) and \(q\), and hence determine the exact value of \(| p - 2 | - | q - 1 |\).

4

1. Sketch, on the same diagram, the graphs of \(y = | 3 - x |\) and \(y = 9 - 2 x\).
2. Solve the inequality \(| 3 - x | > 9 - 2 x\).
3. Use logarithms to solve the inequality \(2 ^ { 3 x - 10 } < 500\). Give your answer in the form \(x < a\), where the value of \(a\) is given correct to 3 significant figures.
4. List the integers that satisfy both of the inequalities \(| 3 - x | > 9 - 2 x\) and \(2 ^ { 3 x - 10 } < 500\).

2 Solve the inequality \(| x - 7 | > 4 x + 3\).

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

1 Solve the inequality \(| x - 4 | > | x + 1 |\).

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

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

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

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

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

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