Inequalities

9 Solve the inequality \(x ( x - 6 ) > 0\).

1 State the set of values of \(x\) which satisfies the inequality $$( x - 3 ) ( 2 x + 7 ) > 0$$ Tick ( \(\checkmark\) ) one box. $$\begin{aligned} & \left\{ x : - \frac { 7 } { 2 } < x < 3 \right\} \\ & \left\{ x : x < - 3 \text { or } x > \frac { 7 } { 2 } \right\} \\ & \left\{ x : x < - \frac { 7 } { 2 } \text { or } x > 3 \right\} \\ & \left\{ x : - 3 < x < \frac { 7 } { 2 } \right\} \end{aligned}$$

14 The inequality $$\left( x ^ { 2 } - 5 x - 24 \right) \left( x ^ { 2 } + 7 x + a \right) < 0$$ has the solution set $$\{ x : - 9 < x < - 3 \} \cup \{ x : 2 < x < b \}$$ Find the values of integers \(a\) and \(b\) \includegraphics[max width=\textwidth, alt={}]{b37e2ee7-1cde-4d75-895a-381b32f4e95a-21_2491_1755_173_123} number Additional page, if required. Write the question numbers in the left-hand margin. \(\_\_\_\_\) number \section*{Additional page, if required. Write the question numbers in the left-hand margin.
Additional page, if required. uestion numbers in the left-hand margin.} \begin{center} \begin{tabular}{|l|l|} \hline Question number & Additional page, if required. Write the question numbers in the left-hand margin.
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4 Solve the inequality \(\frac { 3 } { x - 4 } > 1\).

1. (a) Sketch the curve with equation

$$y = \frac { k } { x } \quad x \neq 0$$ where \(k\) is a positive constant.
(b) Hence or otherwise, solve $$\frac { 16 } { x } \leqslant 2$$

1. In this question you must show all stages of your working.

\section*{Solutions relying on calculator technology are not acceptable.} Given that $$\frac { x + 2 } { x + 4 } \leqslant \frac { x } { k ( x - 1 ) }$$ where \(k\) is a positive constant,

1. show that $$( x + 4 ) ( x - 1 ) \left( p x ^ { 2 } + q x + r \right) \leqslant 0$$ where \(p , q\) and \(r\) are expressions in terms of \(k\) to be determined.
2. Hence, or otherwise, determine the values for \(x\) for which $$\frac { x + 2 } { x + 4 } \leqslant \frac { x } { 3 ( x - 1 ) }$$

6 Solve the inequality \(5 - 2 x < 0\).

1 Solve the inequality \(1 - 2 x < 4 + 3 x\).

1. a) Solve the inequality:

$$\frac { x - 9 } { 2012 } + \frac { x - 8 } { 2013 } + \frac { x - 7 } { 2014 } + \frac { x - 6 } { 2015 } + \frac { x - 5 } { 2016 } \leq \frac { x - 2012 } { 9 } + \frac { x - 2013 } { 8 } + \frac { x - 2014 } { 7 } + \frac { x - 2015 } { 6 } + \frac { x - 2016 } { 5 }$$ b) Find all ( \(x , y , z\) ) such that: $$\frac { 1 } { x } + \frac { 1 } { y + z } = \frac { 1 } { 3 } , \quad \frac { 1 } { y } + \frac { 1 } { z + x } = \frac { 1 } { 5 } , \quad \frac { 1 } { z } + \frac { 1 } { x + y } = \frac { 1 } { 7 }$$ [Question 1 - Continued]
[0pt] [Question 1 - Continued]

1 Find the set of values of \(k\) for which the curve \(y = k x ^ { 2 } - 3 x\) and the line \(y = x - k\) do not meet.

4 A line has equation \(y = 3 x + k\) and a curve has equation \(y = x ^ { 2 } + k x + 6\), where \(k\) is a constant. Find the set of values of \(k\) for which the line and curve have two distinct points of intersection.

2 Find the set of values of \(a\) for which the curve \(y = - \frac { 2 } { x }\) and the straight line \(y = a x + 3 a\) meet at two distinct points.

8 Find the set of values of \(k\) for which the equation \(2 x ^ { 2 } + k x + 2 = 0\) has no real roots.

3

1. Express \(4 x ^ { 2 } - 24 x + p\) in the form \(a ( x + b ) ^ { 2 } + c\), where \(a\) and \(b\) are integers and \(c\) is to be given in terms of the constant \(p\).
2. Hence or otherwise find the set of values of \(p\) for which the equation \(4 x ^ { 2 } - 24 x + p = 0\) has no real roots.

8 The quadratic equation \(k x ^ { 2 } + ( 3 k - 1 ) x - 4 = 0\) has no real roots. Find the set of possible values of \(k\).

1 Show in a sketch the region of the \(x - y\) plane within which all three of the following inequalities hold. \(y \geqslant x ^ { 2 } , \quad x + y \leqslant 2 , \quad x \geqslant 0\). You should indicate the region for which the inequalities hold by labelling the region \(R\).

5. (a) On the same axes, sketch the graphs of \(y = x + 2\) and \(y = x ^ { 2 } - x - 6\) showing the coordinates of all points at which each graph crosses the coordinate axes.
(b) On your sketch, show, by shading, the region \(R\) defined by the inequalities $$y < x + 2 \text { and } y > x ^ { 2 } - x - 6$$ (c) Hence, or otherwise, find the set of values of \(x\) for which \(x ^ { 2 } - 2 x - 8 < 0\) \includegraphics[max width=\textwidth, alt={}, center]{2217be5e-8edd-413f-9c97-212e585ff58d-10_921_1287_699_260} \includegraphics[max width=\textwidth, alt={}, center]{2217be5e-8edd-413f-9c97-212e585ff58d-11_2260_48_313_37} Quadratic: \(y = x ^ { 2 } - x - 6 = ( x - 3 ) ( x + 2 )\) $$x = 3 , \quad x = - 2 @ y = 0$$ Linear: \(\quad y = x + 2\) $$\begin{array} { l l } x = 0 : & y = 2 \quad ( 0,2 ) \\ y = 0 : & x = - 2 \quad ( - 2,0 ) \end{array}$$ c) \(\quad x ^ { 2 } - 2 x - 8 < 0\) $$\begin{aligned} \therefore ( x - 4 ) ( x + 2 ) & < 0 \\ x = 4 \quad x = - 2 & \\ \therefore - 2 < x < 4 & < 4 \end{aligned}$$

9 The diagram below shows the graphs of \(y = x ^ { 2 } - 4 x - 12\) and \(y = x + 2\) \includegraphics[max width=\textwidth, alt={}, center]{11168e8f-5ba5-4d27-83ab-0327cc23d08c-10_933_912_358_566} 9

1. Write down three inequalities which together describe the shaded region.
   9
2. Find the coordinates of the points \(A , B\) and \(C\).
   9
3. Find the exact area of the shaded region.
   Fully justify your answer.
   [0pt] [6 marks]

7 Solve the inequalities

1. \(- 9 \leqslant 6 x + 5 \leqslant 0\),
2. \(6 x + 5 < x ^ { 2 } + 2 x - 7\).

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

1. \(3 ( 2 x + 1 ) > 5 - 2 x\),
2. \(2 x ^ { 2 } - 7 x + 3 > 0\),
3. both \(3 ( 2 x + 1 ) > 5 - 2 x\) and \(2 x ^ { 2 } - 7 x + 3 > 0\).

3 It is given that the variable \(x\) is such that $$1.3 ^ { 2 x } < 80 \quad \text { and } \quad | 3 x - 1 | > | 3 x - 10 | .$$ Find the set of possible values of \(x\), giving your answer in the form \(a < x < b\) where the constants \(a\) and \(b\) are correct to 3 significant figures.

4. The width of a rectangular sports pitch is \(x\) metres, \(x > 0\). The length of the pitch is 20 m more than its width. Given that the perimeter of the pitch must be less than 300 m ,

1. form a linear inequality in \(x\). Given that the area of the pitch must be greater than \(4800 \mathrm {~m} ^ { 2 }\),
2. form a quadratic inequality in \(x\).
3. by solving your inequalities, find the set of possible values of \(x\).

3. \begin{figure}[h] \end{figure} Figure 1 shows the plan of a garden. The marked angles are right angles.
The six edges are straight lines.
The lengths shown in the diagram are given in metres. Given that the perimeter of the garden is greater than 29 m ,

1. show that \(x > 1.5 \mathrm {~m}\) Given also that the area of the garden is less than \(72 \mathrm {~m} ^ { 2 }\),
2. form and solve a quadratic inequality in \(x\).
3. Hence state the range of possible values of \(x\).
   \href{http://www.dynamicpapers.com}{www.dynamicpapers.com}

11 A lawn is to be made in the shape shown below. The units are metres. \includegraphics[max width=\textwidth, alt={}, center]{918d83c3-1608-4482-9d3d-8af05e65f353-4_412_698_486_726}

1. The perimeter of the lawn is \(P \mathrm {~m}\). Find \(P\) in terms of \(x\).
2. Show that the area, \(A \mathrm {~m} ^ { 2 }\), of the lawn is given by \(A = 9 x ^ { 2 } + 6 x\). The perimeter of the lawn must be at least 39 m and the area of the lawn must be less than \(99 \mathrm {~m} ^ { 2 }\).
3. By writing down and solving appropriate inequalities, determine the set of possible values of \(x\).

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

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

1. A physics student is studying the movement of particles in an electric field. In one experiment, the distances in micrometres of two moving particles, \(A\) and \(B\), from a fixed point \(O\) are modelled by

$$\begin{aligned} & d _ { A } = | 5 t - 31 | \\ & d _ { B } = \left| 3 t ^ { 2 } - 25 t + 8 \right| \end{aligned}$$ respectively, where \(t\) is the time in seconds after motion begins.

1. Use algebra to find the range of time for which particle \(A\) is further away from \(O\) than particle \(B\) is from \(O\). It was recorded that the distance of particle \(B\) from \(O\) was less than the distance of particle \(A\) from \(O\) for approximately 4 seconds.
2. Use this information to assess the validity of the model.

2

1. Express \(4 x ^ { 2 } - 12 x\) in the form \(( 2 x + a ) ^ { 2 } + b\).
2. Hence, or otherwise, find the set of values of \(x\) satisfying \(4 x ^ { 2 } - 12 x > 7\).

5 Solve the inequalities

1. \(1 < 4 x - 9 < 5\),
2. \(y ^ { 2 } \geqslant 4 y + 5\).

2 Given that \(y = \frac { x ^ { 2 } + x + 1 } { ( x - 1 ) ^ { 2 } }\), prove that \(y \geqslant \frac { 1 } { 4 }\) for all \(x \neq 1\).

6 Find the positive integer values of \(x\) for which $$\frac { 1 } { 2 } ( 26 - 2 x ) \geq 2 ( 3 + x )$$

1 Solve the inequality \(( 0.8 ) ^ { x } < 0.5\).

7. \begin{figure}[h] \end{figure} The region \(R _ { 1 }\), shown shaded in Figure 2, is defined by the inequalities $$0 \leqslant y \leqslant 2 \quad y \leqslant 10 - 2 x \quad y \leqslant k x$$ where \(k\) is a constant.
The line \(x = a\), where \(a\) is a constant, passes through the intersection of the lines \(y = 2\) and \(y = k x\) Given that the area of \(R _ { 1 }\) is \(\frac { 27 } { 4 }\) square units,

1. find
   1. the value of \(a\)
   2. the value of \(k\)
2. Define the region \(R _ { 2 }\), also shown shaded in Figure 2, using inequalities.

4. (i) Solve the inequality $$x ^ { 2 } - 13 x + 30 < 0$$ (ii) Hence find the set of values of \(y\) such that $$2 ^ { 2 y } - 13 \left( 2 ^ { y } \right) + 30 < 0 .$$

6 A curve has equation \(y = \mathrm { f } ( x )\). It is given that \(\mathrm { f } ^ { \prime } ( x ) = 3 x ^ { 2 } + 2 x - 5\).

1. Find the set of values of \(x\) for which f is an increasing function.
2. Given that the curve passes through \(( 1,3 )\), find \(\mathrm { f } ( x )\).

6 A curve has equation \(y = x ^ { 2 } - x + 3\) and a line has equation \(y = 3 x + a\), where \(a\) is a constant.

1. Show that the \(x\)-coordinates of the points of intersection of the line and the curve are given by the equation \(x ^ { 2 } - 4 x + ( 3 - a ) = 0\).
2. For the case where the line intersects the curve at two points, it is given that the \(x\)-coordinate of one of the points of intersection is - 1 . Find the \(x\)-coordinate of the other point of intersection.
3. For the case where the line is a tangent to the curve at a point \(P\), find the value of \(a\) and the coordinates of \(P\).

9 A curve has equation \(y = 2 x ^ { 2 } - 3 x + 1\) and a line has equation \(y = k x + k ^ { 2 }\), where \(k\) is a constant.

1. Show that, for all values of \(k\), the curve and the line meet.
2. State the value of \(k\) for which the line is a tangent to the curve and find the coordinates of the point where the line touches the curve.

2 The function f is defined for \(x \in \mathbb { R }\) by \(\mathrm { f } ( x ) = x ^ { 2 } - 6 x + c\), where \(c\) is a constant. It is given that \(\mathrm { f } ( x ) > 2\) for all values of \(x\). Find the set of possible values of \(c\).

3 Consider the following linear programming problem:
Maximise \(\quad 3 x + 4 y\) subject to \(\quad 2 x + 5 y \leqslant 60\) \(x + 2 y \leqslant 25\) \(x + y \leqslant 18\)

1. Graph the inequalities and hence solve the LP.
2. The right-hand side of the second inequality is increased from 25 . At what new value will this inequality become redundant?