1.02g Inequalities: linear and quadratic in single variable

8 A curve has equation \(y = \frac { 1 } { x ^ { 2 } - 4 }\).

1. 1. Write down the equations of the three asymptotes of the curve.
   2. Sketch the curve, showing the coordinates of any points of intersection with the coordinate axes.
2. Hence, or otherwise, solve the inequality $$\frac { 1 } { x ^ { 2 } - 4 } < - 2$$

5 The curve \(C\) has equation \(y = \frac { x } { ( x + 1 ) ( x - 2 ) }\).
The line \(L\) has equation \(y = - \frac { 1 } { 2 }\).

1. Write down the equations of the asymptotes of \(C\).
2. The line \(L\) intersects the curve \(C\) at two points. Find the \(x\)-coordinates of these two points.
3. Sketch \(C\) and \(L\) on the same axes.
   (You are given that the curve \(C\) has no stationary points.)
4. Solve the inequality $$\frac { x } { ( x + 1 ) ( x - 2 ) } \leqslant - \frac { 1 } { 2 }$$

9 A curve has equation $$y = \frac { x ^ { 2 } - 2 x + 1 } { x ^ { 2 } - 2 x - 3 }$$

1. Find the equations of the three asymptotes of the curve.
2. 1. Show that if the line \(y = k\) intersects the curve then $$( k - 1 ) x ^ { 2 } - 2 ( k - 1 ) x - ( 3 k + 1 ) = 0$$
   2. Given that the equation \(( k - 1 ) x ^ { 2 } - 2 ( k - 1 ) x - ( 3 k + 1 ) = 0\) has real roots, show that $$k ^ { 2 } - k \geqslant 0$$
   3. Hence show that the curve has only one stationary point and find its coordinates.
      (No credit will be given for solutions based on differentiation.)
3. Sketch the curve and its asymptotes.

8 A curve \(C\) has equation $$y = \frac { x ( x - 3 ) } { x ^ { 2 } + 3 }$$

1. State the equation of the asymptote of \(C\).
2. The line \(y = k\) intersects the curve \(C\). Show that \(4 k ^ { 2 } - 4 k - 3 \leqslant 0\).
3. Hence find the coordinates of the stationary points of the curve \(C\). (No credit will be given for solutions based on differentiation.) \includegraphics[max width=\textwidth, alt={}, center]{e45b07a3-e303-4caf-8f3a-5341bad7560a-24_2488_1728_219_141}

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

$$\frac { x } { x ^ { 2 } - 2 x - 3 } \leqslant \frac { 1 } { x + 3 }$$

1. A student was set the following problem.

Use algebra to find the set of values of \(x\) for which $$\frac { x } { x - 24 } > \frac { 1 } { x + 11 }$$ The student's attempt at a solution is written below. $$\begin{gathered} x ( x - 24 ) ( x + 11 ) ^ { 2 } > ( x + 11 ) ( x - 24 ) ^ { 2 } \\ x ( x - 24 ) ( x + 11 ) ^ { 2 } - ( x + 11 ) ( x - 24 ) ^ { 2 } > 0 \\ ( x - 24 ) ( x + 11 ) [ x ( x + 11 ) - x - 24 ] > 0 \\ ( x - 24 ) ( x + 11 ) \left[ x ^ { 2 } + 10 x - 24 \right] > 0 \\ ( x - 24 ) ( x + 11 ) ( x + 12 ) ( x - 2 ) > 0 \\ x = 24 , x = - 11 , x = - 12 , x = 2 \\ \{ x \in \mathbb { R } : - 12 < x < - 11 \} \cup \{ x \in \mathbb { R } : 2 < x < 24 \} \end{gathered}$$ Line 3 There are errors in the student's solution.

1. Identify the error made
   1. in line 3
   2. in line 7
2. Find a correct solution to this problem.

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

$$\frac { x + 1 } { 2 x ^ { 2 } + 5 x - 3 } > \frac { x } { 4 x ^ { 2 } - 1 }$$

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

$$x ( x - 1 ) > \frac { x - 1 } { x }$$ giving your answer in set notation.

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

$$x \geqslant \frac { 2 x + 15 } { 2 x + 3 }$$

1. (a) Use algebra to determine the values of \(x\) for which

$$\frac { 5 x } { x - 2 } \geqslant 12$$ (b) Hence, given that \(x\) is an integer, deduce the value of \(x\).

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

$$\frac { 1 } { x } < \frac { x } { x + 2 }$$

3. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \mathrm { f } ( x )\) where $$f ( x ) = \frac { x } { | x | - 2 }$$ Use algebra to determine the values of \(x\) for which $$2 x - 5 > \frac { x } { | x | - 2 }$$

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve with equation \(y = \left| x ^ { 2 } - 8 \right|\) and a sketch of the straight line with equation \(y = m x + c\), where \(m\) and \(c\) are positive constants. The equation $$\left| x ^ { 2 } - 8 \right| = m x + c$$ has exactly 3 roots, as shown in Figure 1.

1. Show that $$m ^ { 2 } - 4 c + 32 = 0$$ Given that \(c = 3 m\)
2. determine the value of \(m\) and the value of \(c\)
3. Hence solve $$\left| x ^ { 2 } - 8 \right| \geqslant m x + c$$

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

$$\left| x ^ { 2 } - 2 x \right| \leqslant x$$

1 In this question you must show detailed reasoning. Solve the inequality \(10 x ^ { 2 } + x - 2 > 0\).

3 In this question you must show detailed reasoning. A gardener is planning the design for a rectangular flower bed. The requirements are:

• the length of the flower bed is to be 3 m longer than the width,
• the length of the flower bed must be at least 14.5 m ,
• the area of the flower bed must be less than \(180 \mathrm {~m} ^ { 2 }\).

The width of the flower bed is \(x \mathrm {~m}\).
By writing down and solving appropriate inequalities in \(x\), determine the set of possible values for the width of the flower bed.

8 In this question you must show detailed reasoning.

1. Use differentiation to find the coordinates of the stationary point on the curve with equation \(y = 2 x ^ { 2 } - 3 x - 2\).
2. Use the second derivative to determine the nature of the stationary point.
3. Show by shading on a sketch the region defined by the inequality \(y \geqslant 2 x ^ { 2 } - 3 x - 2\), indicating clearly whether the boundary is included or not.
4. Solve the inequality \(2 x ^ { 2 } - 3 x - 2 > 0\) using set notation for your answer.

13 In this question you must show detailed reasoning. Determine the values of \(k\) for which part of the graph of \(y = x ^ { 2 } - k x + 2 k\) appears below the \(x\)-axis.

1 Solve the following inequalities.

1. \(- 5 < 3 x + 1 < 14\)
2. \(4 x ^ { 2 } + 3 > 28\)

2 \includegraphics[max width=\textwidth, alt={}, center]{a16ab26f-21fb-4a73-8b94-c16bef611bcb-4_661_579_831_246} The diagram shows a patio. The perimeter of the patio has to be less than 44 m .
The area of the patio has to be at least \(45 \mathrm {~m} ^ { 2 }\).

1. Write down, in terms of \(x\), an inequality satisfied by
   1. the perimeter of the patio,
   2. the area of the patio.
2. Hence determine the set of possible values of \(x\).

7

1. Sketch the curve \(y = 2 x ^ { 2 } - x - 3\).
2. Hence, or otherwise, solve \(2 x ^ { 2 } - x - 3 < 0\).
3. Given that the equation \(2 x ^ { 2 } - x - 3 = k\) has no real roots, find the set of possible values of k .

7 The curve \(C\) has equation \(y = x ^ { 2 } + 7\). The line \(L\) has equation \(y = k ( 3 x + 1 )\), where \(k\) is a constant.

1. Show that the \(x\)-coordinates of any points of intersection of the line \(L\) with the curve \(C\) satisfy the equation $$x ^ { 2 } - 3 k x + 7 - k = 0$$
2. The curve \(C\) and the line \(L\) intersect in two distinct points. Show that $$9 k ^ { 2 } + 4 k - 28 > 0$$
3. Solve the inequality \(9 k ^ { 2 } + 4 k - 28 > 0\).

7 The quadratic equation $$( 2 k - 3 ) x ^ { 2 } + 2 x + ( k - 1 ) = 0$$ where \(k\) is a constant, has real roots.

1. Show that \(2 k ^ { 2 } - 5 k + 2 \leqslant 0\).
2. 1. Factorise \(2 k ^ { 2 } - 5 k + 2\).
   2. Hence, or otherwise, solve the quadratic inequality $$2 k ^ { 2 } - 5 k + 2 \leqslant 0$$