Solve |f(x)| compared to |g(x)| with parameters

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 Solve the equation \(| x + a | = | 2 x - 5 a |\), giving \(x\) in terms of the positive constant \(a\).

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

1 Find the set of values of \(x\) satisfying the inequality $$| x + 2 a | > 3 | x - a |$$ where \(a\) is a positive constant.

1 Find the set of values of \(x\) satisfying the inequality \(2 | 2 x - a | < | x + 3 a |\), where \(a\) is a positive constant. [4]

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

6. Given that \(a\) and \(b\) are constants and that \(a > b > 0\)

1. on separate diagrams, sketch the graph with equation
   1. \(y = | x - a |\)
   2. \(y = | x - a | - b\) Show on each sketch the coordinates of each point at which the graph crosses or meets the \(x\)-axis and the \(y\)-axis.
2. Hence or otherwise find the complete set of values of \(x\) for which $$| x - a | - b < \frac { 1 } { 2 } x$$ giving your answer in terms of \(a\) and \(b\).

1. Given that \(a\) is a positive constant,
   a. Sketch the graph with equation

$$y = | a - 2 x |$$ Show on your sketch the coordinates of each point at which the graph crosses the \(x\)-axis and \(y\)-axis.
b. Solve the inequality \(| a - 2 x | > x + 2 a\)

11. \begin{figure}[h] \end{figure} Figure 4 shows a sketch of the graph with equation $$y = | 2 x - 3 k |$$ where \(k\) is a positive constant.

1. Sketch the graph with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = k - | 2 x - 3 k |$$ stating
   • the coordinates of the maximum point
   • the coordinates of any points where the graph cuts the coordinate axes
   • Find, in terms of \(k\), the set of values of \(x\) for which
   $$k - | 2 x - 3 k | > x - k$$ giving your answer in set notation.
2. Find, in terms of \(k\), the coordinates of the minimum point of the graph with equation $$y = 3 - 5 f \left( \frac { 1 } { 2 } x \right)$$

3. The function \(f\) is defined by $$f : x \text { a } | 2 x - a | , \quad x \in ^ { \circ }$$ where \(a\) is a positive constant.

1. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of the points where the graph cuts the axes.
2. On a separate diagram, sketch the graph of \(y = \mathrm { f } ( 2 x )\), showing the coordinates of the points where the graph cuts the axes.
3. Given that a solution of the equation \(\mathrm { f } ( x ) = \frac { 1 } { 2 } x\) is \(x = 4\), find the two possible values of \(a\).

5. The functions \(f\) and \(g\) are defined by $$\begin{aligned} & \mathrm { f } : x \alpha \quad | x - a | + a , x \in \mathbb { R } \\ & \mathrm {~g} : x \alpha \quad 4 x + a , \quad x \in \mathbb { R } \end{aligned}$$ where \(a\) is a positive constant.

1. On the same diagram, sketch the graphs of f and g , showing clearly the coordinates of any points at which your graphs meet the axes.
2. Use algebra to find, in terms of \(a\), the coordinates of the point at which the graphs of f and g intersect.
3. Find an expression for \(\mathrm { fg } ( x )\).
4. Solve, for \(x\) in terms of \(a\), the equation $$\mathrm { fg } ( x ) = 3 a$$ \section*{6.}

3. (a) Sketch the graph of \(y = | 2 x + a | , a > 0\), showing the coordinates of the points where the graph meets the coordinate axes.
(b) On the same axes, sketch the graph of \(y = \frac { 1 } { x }\).
(c) Explain how your graphs show that there is only one solution of the equation $$x | 2 x + a | - 1 = 0$$ (d) Find, using algebra, the value of \(x\) for which \(x | 2 x + 1 | - 1 = 0\).

7. The function \(f\) is defined by $$f : x \wp \rightarrow | 2 x - a | , x \in \mathbb { R }$$ where \(a\) is a positive constant.

1. Sketch the graph of \(y = \mathrm { f } ( x )\), showing the coordinates of the points where the graph cuts the axes.
2. On a separate diagram, sketch the graph of \(y = \mathrm { f } ( 2 x )\), showing the coordinates of the points where the graph cuts the axes.
3. Given that a solution of the equation \(\mathrm { f } ( x ) = \frac { 1 } { 2 } x\) is \(x = 4\), find the two possible values of \(a\).

7. (a) Sketch the graph of \(y = \left| x ^ { 2 } - a ^ { 2 } \right|\), where \(a > 1\), showing the coordinates of the points where the graph meets the axes.
(b) Solve \(\left| x ^ { 2 } - a ^ { 2 } \right| = a ^ { 2 } - x , a > 1\).
(c) Find the set of values of \(x\) for which \(\left| x ^ { 2 } - a ^ { 2 } \right| > a ^ { 2 } - x , a > 1\).