1.02g Inequalities: linear and quadratic in single variable

9 Functions f and g are defined by $$\begin{aligned} & \mathrm { f } ( x ) = 2 x ^ { 2 } + 8 x + 1 \quad \text { for } x \in \mathbb { R } \\ & \mathrm {~g} ( x ) = 2 x - k \quad \text { for } x \in \mathbb { R } \end{aligned}$$ where \(k\) is a constant.

1. Find the value of \(k\) for which the line \(y = \mathrm { g } ( x )\) is a tangent to the curve \(y = \mathrm { f } ( x )\).
2. In the case where \(k = - 9\), find the set of values of \(x\) for which \(\mathrm { f } ( x ) < \mathrm { g } ( x )\).
3. In the case where \(k = - 1\), find \(\mathrm { g } ^ { - 1 } \mathrm { f } ( x )\) and solve the equation \(\mathrm { g } ^ { - 1 } \mathrm { f } ( x ) = 0\).
4. Express \(\mathrm { f } ( x )\) in the form \(2 ( x + a ) ^ { 2 } + b\), where \(a\) and \(b\) are constants, and hence state the least value of \(\mathrm { f } ( x )\).

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 \(3 x + 5 < | 2 x - 3 |\).

3

1. Sketch, on a single diagram, the graphs of \(y = \left| \frac { 1 } { 2 } x - a \right|\) and \(y = \frac { 3 } { 2 } x - \frac { 1 } { 2 } a\), where \(a\) is a positive constant.
2. Find the coordinates of the point of intersection of the two graphs.
3. Deduce the solution of the inequality \(\left| \frac { 1 } { 2 } x - a \right| > \frac { 3 } { 2 } x - \frac { 1 } { 2 } a\).

2

1. Sketch, on the same diagram, the graphs of \(y = 3 x\) and \(y = | x - 3 |\).
2. Find the coordinates of the point where the two graphs intersect.
3. Deduce the solution of the inequality \(3 x < | x - 3 |\).

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

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 \(2 ^ { 3 x - 10 } < 500\).

4 The polynomial \(x ^ { 4 } - 2 x ^ { 3 } - 2 x ^ { 2 } + a\) is denoted by \(\mathrm { f } ( x )\). It is given that \(\mathrm { f } ( x )\) is divisible by \(x ^ { 2 } - 4 x + 4\).

1. Find the value of \(a\).
2. When \(a\) has this value, show that \(\mathrm { f } ( x )\) is never negative.

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

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

4 The polynomial \(4 x ^ { 3 } + a x + 2\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that ( \(2 x + 1\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\).
2. When \(a\) has this value,
   1. factorise \(\mathrm { p } ( x )\),
   2. solve the inequality \(\mathrm { p } ( x ) > 0\), justifying your answer.

1 Solve the inequality \(| 9 - 2 x | < 1\).

3 The polynomial \(2 x ^ { 3 } + a x ^ { 2 } - 4\) is denoted by \(\mathrm { p } ( x )\). It is given that ( \(x - 2\) ) is a factor of \(\mathrm { p } ( x )\).

1. Find the value of \(a\). When \(a\) has this value,
2. factorise \(\mathrm { p } ( x )\),
3. solve the inequality \(\mathrm { p } ( x ) > 0\), justifying your answer.

5 The polynomial \(4 x ^ { 3 } - 4 x ^ { 2 } + 3 x + a\), where \(a\) is a constant, is denoted by \(\mathrm { p } ( x )\). It is given that \(\mathrm { p } ( x )\) is divisible by \(2 x ^ { 2 } - 3 x + 3\).

1. Find the value of \(a\).
2. When \(a\) has this value, solve the inequality \(\mathrm { p } ( x ) < 0\), justifying your answer.

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

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

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 Let \(a\) be a positive constant.

1. Sketch the curve with equation \(\mathrm { y } = \frac { \mathrm { ax } } { \mathrm { x } + 7 }\).
2. Sketch the curve with equation \(y = \left| \frac { a x } { x + 7 } \right|\) and find the set of values of \(x\) for which \(\left| \frac { a x } { x + 7 } \right| > \frac { a } { 2 }\).

6 The curve \(C\) has equation \(\mathrm { y } = \frac { 10 + \mathrm { x } - 2 \mathrm { x } ^ { 2 } } { 2 \mathrm { x } - 3 }\).

1. Find the equations of the asymptotes of \(C\).
2. Show that \(C\) has no turning points.
3. Sketch \(C\), stating the coordinates of the intersections with the axes.
4. Sketch the curve with equation \(y = \left| \frac { 10 + x - 2 x ^ { 2 } } { 2 x - 3 } \right|\) and find in exact form the set of values of \(x\) for which \(\left| \frac { 10 + x - 2 x ^ { 2 } } { 2 x - 3 } \right| < 4\).

7 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + \mathrm { x } + 9 } { \mathrm { x } + 1 }\).

1. Find the equations of the asymptotes of \(C\).
2. Find the coordinates of the stationary points on \(C\).
3. Sketch \(C\), stating the coordinates of any intersections with the axes.
4. Sketch the curve with equation \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } + \mathrm { x } + 9 } { \mathrm { x } + 1 } \right|\) and find the set of values of \(x\) for which \(2 \left| x ^ { 2 } + x + 9 \right| > 13 | x + 1 |\). If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.

6 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + 2 \mathrm { x } - 15 } { \mathrm { x } - 2 }\).

1. Find the equations of the asymptotes of \(C\).
2. Show that \(C\) has no stationary points.
3. Sketch \(C\), stating the coordinates of the intersections with the axes.
4. Sketch the curve with equation \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } - 2 \mathrm { x } - 15 } { \mathrm { x } - 2 } \right|\).
5. Find the set of values of \(x\) for which \(\left| \frac { 2 x ^ { 2 } + 4 x - 30 } { x - 2 } \right| < 15\).

6 The curve \(C\) has equation \(\mathrm { y } = \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 }\), where \(a > \frac { 5 } { 2 }\).

1. Find the equations of the asymptotes of \(C\).
2. Show that \(C\) has no stationary points.
3. Sketch \(C\), stating the coordinates of the point of intersection with the \(y\)-axis and labelling the asymptotes.
4. 1. Sketch the curve with equation \(\mathrm { y } = \left| \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 } \right|\).
   2. On your sketch in part (i), draw the line \(\mathrm { y } = \mathrm { a }\).
   3. It is given that \(\left| \frac { \mathrm { x } ^ { 2 } + \mathrm { ax } + 1 } { \mathrm { x } + 2 } \right| < \mathrm { a }\) for \(- 5 - \sqrt { 14 } < x < - 3\) and \(- 5 + \sqrt { 14 } < x < 3\). Find the value of \(a\).

3

1. Solve the inequality \(| 2 x - 5 | < | x + 3 |\).
2. Hence find the largest integer \(y\) satisfying the inequality \(| 2 \ln y - 5 | < | \ln y + 3 |\).

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

3

1. Solve the inequality \(| y - 5 | < 1\).
2. Hence solve the inequality \(\left| 3 ^ { x } - 5 \right| < 1\), giving 3 significant figures in your answer.