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

3. Use algebra to obtain the set of values of \(x\) for which $$\left| \frac { x ^ { 2 } + 3 x + 10 } { x + 2 } \right| < 7 - x$$

1. Use algebra to find the set of values of \(x\) for which

$$\left| 2 x ^ { 2 } + x - 3 \right| > 3 ( 1 - x )$$ [Solutions based entirely on graphical or numerical methods are not acceptable.] \includegraphics[max width=\textwidth, alt={}, center]{0d44aec7-a6e8-47fc-a215-7c8c4790e93f-21_2647_1840_118_111}

1. Find the set of values for which

$$| x - 1 | > 6 x - 1$$

10.

1. Sketch, on the same axes, the graphs with equation \(y = | 2 x - 3 |\), and the line with equation \(y = 5 x - 1\).
2. Solve the inequality \(| 2 x - 3 | < 5 x - 1\).
   

3.

1. Sketch, on the same axes, the graph of \(y = | ( x - 2 ) ( x - 4 ) |\), and the line with equation \(y = 6 - 2 x\).
2. Find the exact values of \(x\) for which \(| ( x - 2 ) ( x - 4 ) | = 6 - 2 x\).
3. Hence solve the inequality \(| ( x - 2 ) ( x - 4 ) | < 6 - 2 x\).
   (2)(Total 11 marks)
   

6. Find the complete set of values of \(x\) for which $$\left| x ^ { 2 } - 2 \right| > 2 x$$

6.

1. On the same diagram, sketch the graphs of \(y = \left| x ^ { 2 } - 4 \right|\) and \(y = | 2 x - 1 |\), showing the coordinates of the points where the graphs meet the axes.
2. Solve \(\left| x ^ { 2 } - 4 \right| = | 2 x - 1 |\), giving your answers in surd form where appropriate.
3. Hence, or otherwise, find the set of values of \(x\) for which \(\left| x ^ { 2 } - 4 \right| > | 2 x - 1 |\).
   (3)(Total 12 marks)
   

3.

1. Use algebra to find the exact solutions of the equation $$\left| 2 x ^ { 2 } + x - 6 \right| = 6 - 3 x$$
2. On the same diagram, sketch the curve with equation \(y = \left| 2 x ^ { 2 } + x - 6 \right|\) and the line with equation \(y = 6 - 3 x\).
3. Find the set of values of \(x\) for which $$\left| 2 x ^ { 2 } + x - 6 \right| > 6 - 3 x$$ (3)(Total 12 marks)
   

2. \includegraphics[max width=\textwidth, alt={}, center]{d6befd60-de40-41b6-8ae5-48656dbca40c-1_734_1228_888_479} The diagram above shows a sketch of the curve with equation $$y = \frac { x ^ { 2 } - 1 } { | x + 2 | } , \quad x \neq - 2$$ The curve crosses the \(x\)-axis at \(x = 1\) and \(x = - 1\) and the line \(x = - 2\) is an asymptote of the curve.

1. Use algebra to solve the equation \(\frac { x ^ { 2 } - 1 } { | x + 2 | } = 3 ( 1 - x )\).
2. Hence, or otherwise, find the set of values of \(x\) for which $$\frac { x ^ { 2 } - 1 } { | x + 2 | } < 3 ( 1 - x )$$ (Total 9 marks)

3.

1. Find the set of values of \(x\) for which $$x + 4 > \frac { 2 } { x + 3 }$$
2. Deduce, or otherwise find, the values of \(x\) for which $$x + 4 > \frac { 2 } { | x + 3 | }$$

1. Find the set of values of \(x\) for which

$$\left| x ^ { 2 } - 4 \right| > 3 x$$

2. Use algebra to find the set of values of \(x\) for which $$\left| 3 x ^ { 2 } - 19 x + 20 \right| < 2 x + 2$$

4. Use algebra to find the set of values of \(x\) for which $$\left| x ^ { 2 } - 2 \right| > 4 x$$

Figure 1 shows the graph of equation \(y = \mathrm { f } ( x )\).
The points \(P ( - 3,0 )\) and \(Q ( 2 , - 4 )\) are stationary points on the graph.
Sketch, on separate diagrams, the graphs of

1. \(y = 3 \mathrm { f } ( x + 2 )\)
2. \(y = | \mathrm { f } ( x ) |\) On each diagram, show the coordinates of any stationary points.

\begin{figure}[h] \end{figure} Figure 1 shows part of the curve with equation \(y = \mathrm { f } ( x ) , x \in \mathbb { R }\), where f is an increasing function of \(x\). The curve passes through the points \(P ( 0 , - 2 )\) and \(Q ( 3,0 )\) as shown. In separate diagrams, sketch the curve with equation

1. \(y = | f ( x ) |\),
2. \(y = \mathrm { f } ^ { - 1 } ( x )\),
3. \(y = \frac { 1 } { 2 } \mathrm { f } ( 3 x )\). Indicate clearly on each sketch the coordinates of the points at which the curve crosses or meets the axes.

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

8. The functions f and g are defined for all real values of \(x\) by $$\begin{aligned} & \mathrm { f } : x \rightarrow | x - 3 a | \\ & \mathrm { g } : x \rightarrow | 2 x + a | \end{aligned}$$ where \(a\) is a positive constant.

1. Evaluate fg(-2a).
2. Sketch on the same diagram the graphs of \(y = \mathrm { f } ( x )\) and \(y = \mathrm { g } ( x )\), showing the coordinates of any points where each graph meets the coordinate axes.
3. Solve the equation $$\mathrm { f } ( x ) = \mathrm { g } ( x )$$

1. Find the set of values of \(x\) such that

$$| 2 x - 3 | > | x + 2 |$$