Write inequalities from graph

3. (i) Solve
(ii) $$\frac { 3 } { x } > 4$$ Figure 1 shows a sketch of the curve \(C\) and the straight line \(l\).
The infinite region \(R\), shown shaded in Figure 1, lies in quadrants 2 and 3 and is bounded by \(C\) and \(l\) only. Given that

• \(\quad l\) has a gradient of 3
• \(C\) has equation \(y = 2 x ^ { 2 } - 50\)
• \(\quad C\) and \(l\) intersect on the negative \(x\)-axis
  use inequalities to define the region \(R\).

\begin{figure}[h] \end{figure} Figure 1 shows a sketch of the curve \(C\) and the straight line \(l\).
The infinite region \(R\), shown shaded in Figure 1, lies in quadrants 2 and 3 and is bounded by \(C\) and \(l\) only.
Given that

• \(l\) has a gradient of 3
• \(C\) has equation \(y = 2 x ^ { 2 } - 50\)
• \(C\) and \(l\) intersect on the negative \(x\)-axis
  use inequalities to define 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) \(\quad x ^ { 2 } - 2 x - 8 < 0\) $$\begin{aligned} \therefore ( x - 4 ) ( x + 2 ) & < 0 \\ x = 4 \quad x = - 2 & \\ \therefore - 2 < x < 4 & < 4 \end{aligned}$$

8. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of a curve \(C\) and a straight line \(l\).
Given that

• \(C\) has equation \(y = \mathrm { f } ( x )\) where \(\mathrm { f } ( x )\) 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.

12. \begin{figure}[h] \end{figure} Figure 3 shows a sketch of the curve \(C\) with equation \(y = 3 x - 2 \sqrt { x } , x \geqslant 0\) and the line \(l\) with equation \(y = 8 x - 16\) The line cuts the curve at point \(A\) as shown in Figure 3.

1. Using algebra, find the \(x\) coordinate of point \(A\).
2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{fa7abe9f-f5c0-4578-afd1-73176c717536-24_636_780_1585_644} \captionsetup{labelformat=empty} \caption{Figure 4}
   \end{figure} The region \(R\) is shown unshaded in Figure 4. Identify the inequalities that define \(R\).

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a curve \(C\) with equation \(y = \mathrm { f } ( x )\) and a straight line \(l\). The curve \(C\) meets \(l\) at the points \(( 2,4 )\) and \(( 6,0 )\) as shown. The shaded region \(R\), shown shaded in Figure 1, is bounded by \(\mathrm { C } , l\) and the \(y\)-axis. Given that \(\mathrm { f } ( x )\) is a quadratic function in \(x\), use inequalities to define region \(R\).

7. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a curve \(C\) with equation \(y = \mathrm { f } ( x )\) and a straight line \(l\).
The curve \(C\) meets \(l\) at the points \(( - 2,13 )\) and \(( 0,25 )\) as shown.
The shaded region \(R\) is bounded by \(C\) and \(l\) as shown in Figure 1.
Given that

• \(\mathrm { f } ( x )\) is a quadratic function in \(x\)
• ( \(- 2,13\) ) is the minimum turning point of \(y = \mathrm { f } ( x )\) use inequalities to 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 \(R\).

3

1. The diagram shows the line \(y = x + 5\) and the curve \(y = 8 - 2 x - x ^ { 2 }\). The shaded region is the finite region between the line and the curve. The curved part of the boundary is included in the region but the straight part is not included. Write down the inequalities that define the shaded region. \includegraphics[max width=\textwidth, alt={}, center]{4fac72cb-85cb-48d9-8817-899ef3f80a0f-04_846_716_1379_322} \section*{(b) In this question you must show detailed reasoning.} Solve the inequality \(8 - 2 x - x ^ { 2 } > x + 5\) giving your answer in exact form.

11. \includegraphics[max width=\textwidth, alt={}, center]{b1befa4f-5ef6-46e1-afb4-3a3582db7dfd-4_609_951_1541_605} The diagram shows a sketch of the curve \(y = 6 + 4 x - x ^ { 2 }\) and the line \(y = x + 2\). The point \(P\) has coordinates ( \(a , b\) ). Write down the three inequalities involving \(a\) and \(b\) which are such that the point \(P\) will be strictly contained within the shaded area above, if and only if, all three inequalities are satisfied.

1 Show in a sketch the region of the \(x - y\) plane within which all three of the following inequalities are satisfied. $$3 y \geqslant 4 x \quad y - x \leqslant 1 \quad y \geqslant ( x - 1 ) ^ { 2 }$$ You should indicate the region for which the inequalities hold by labelling the region \(R\).

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 \(R\).

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]

4 Sketch the region defined by the inequalities $$y \leq ( 1 - 2 x ) ( x + 3 ) \text { and } y - x \leq 3$$ Clearly indicate your region by shading it in and labelling it \(R\). \includegraphics[max width=\textwidth, alt={}, center]{c8a41c47-bbda-4e91-a7a2-d0bcf6a46f25-03_1000_1004_833_518}

7 The graphs with equations $$y = 2 + 3 x - 2 x ^ { 2 } \text { and } x + y = 1$$ are shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{deec0d32-b031-4227-bc80-7150a0acbc94-10_791_721_550_719} The graphs intersect at the points \(A\) and \(B\) 7

1. On the diagram above, shade and label the region, \(R\), that is satisfied by the inequalities $$0 \leq y \leq 2 + 3 x - 2 x ^ { 2 }$$ and $$x + y \geq 1$$ 7
2. Find the exact coordinates of \(A\)