Inequalities

3 Solve \(3 x ^ { 2 } + 11 x - 20 > 0\).

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.}

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 Solve \(5 x + 3 < | 3 x - 1 |\).

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.

1 Determine the set of values of the constant \(k\) for which the line \(y = 4 x + k\) does not intersect the curve \(y = x ^ { 2 }\).

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. Find the set of values of \(k\) for which the equation \(8 x ^ { 2 } + k x + 2 = 0\) has no real roots.
2. Solve the equation \(8 \cos ^ { 2 } \theta - 10 \cos \theta + 2 = 0\) for \(0 ^ { \circ } \leqslant \theta \leqslant 180 ^ { \circ }\).

9 Find the set of values of \(k\) for which the equation \(x ^ { 2 } + 2 x + 11 = k ( 2 x - 1 )\) has two distinct real roots.

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.

8 The length of a rectangular children's playground is 10 m more than its width. The width of the playground is \(x\) metres.

1. The perimeter of the playground is greater than 64 m . Write down a linear inequality in \(x\).
2. The area of the playground is less than \(299 \mathrm {~m} ^ { 2 }\). Show that \(( x - 13 ) ( x + 23 ) < 0\).
3. By solving the inequalities in parts (i) and (ii), determine 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 \(x\).

2 \includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-03_835_545_749_244} The diagram shows the line \(y = - 2 x + 4\) and the curve \(y = x ^ { 2 } - 4\). The region \(R\) is the unshaded region together with its boundaries. Write down the inequalities that define \(R\).

2 \includegraphics[max width=\textwidth, alt={}, center]{31b0d5b6-1593-489b-bbcd-486e7c96ff18-03_835_545_749_244} The diagram shows the line \(y = - 2 x + 4\) and the curve \(y = x ^ { 2 } - 4\). The region \(R\) is the unshaded region together with its boundaries. Write down the inequalities that define \(R\).

\includegraphics{figure_3} Figure 3 shows a sketch of a curve \(C\) and a straight line \(l\). Given that • \(C\) has equation \(y = f(x)\) where \(f(x)\) is a quadratic expression in \(x\) • \(C\) cuts the \(x\)-axis at \(0\) and \(6\) • \(l\) cuts the \(y\)-axis at \(60\) and intersects \(C\) at the point \((10, 80)\) use inequalities to define the region \(R\) shown shaded in Figure 3. [5]

Sketch the region defined by the inequalities $$y \leq (1 - 2x)(x + 3) \text{ and } y - x \leq 3$$ Clearly indicate your region by shading it in and labelling it \(R\). [3 marks] \includegraphics{figure_4}

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 \(4 x ^ { 2 } - 12 x > 7\).

1 Solve the following inequalities.

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

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

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

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

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 \(c\).

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.

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 \(P\).

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 \(\mathrm { f } ( x )\).

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

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?