1.02g Inequalities: linear and quadratic in single variable

OCR MEI C1 Q14

4 marks Easy -1.8

14 Solve the inequality \(\frac { 3 ( 2 x + 1 ) } { 4 } > - 6\).

OCR MEI C1 Q15

12 marks Moderate -0.3

15

1. Write \(x ^ { 2 } - 5 x + 8\) in the form \(( x - a ) ^ { 2 } + b\) and hence show that \(x ^ { 2 } - 5 x + 8 > 0\) for all values of \(x\).
2. Sketch the graph of \(y = x ^ { 2 } - 5 x + 8\), showing the coordinates of the turning point.
3. Find the set of values of \(x\) for which \(x ^ { 2 } - 5 x + 8 > 14\).
4. If \(\mathrm { f } ( x ) = x ^ { 2 } - 5 x + 8\), does the graph of \(y = \mathrm { f } ( x ) - 10\) cross the \(x\)-axis? Show how you decide.

OCR C2 Q4

6 marks Standard +0.3

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

OCR C3 Q1

5 marks Standard +0.3

1. Find the set of values of \(x\) such that

$$| 2 x - 3 | > | x + 2 |$$

OCR C3 2006 June Q2

5 marks Standard +0.3

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

OCR C3 2007 June Q2

5 marks Standard +0.3

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

OCR MEI C3 Q2

4 marks Moderate -0.8

2

1. Sketch the graph of \(y = | 2 x - 3 |\).
2. Hence, or otherwise, solve the inequality \(| 2 x - 3 | < 5\). Illustrate your answer on your graph.

OCR MEI FP1 2005 January Q3

7 marks Standard +0.3

3

1. Solve the equation \(\frac { 1 } { x + 2 } = 3 x + 4\).
2. Solve the inequality \(\frac { 1 } { x + 2 } \leqslant 3 x + 4\).

OCR MEI FP1 2005 January Q7

14 marks Standard +0.8

7 A curve has equation \(y = \frac { ( 2 x - 3 ) ( x + 1 ) } { ( x + 4 ) ( x - 2 ) }\).

1. Write down the values of \(x\) for which \(y = 0\).
2. Write down the equations of the three asymptotes.
3. Determine whether the curve approaches the horizontal asymptote from above or from below for
   (A) large positive values of \(x\),
   (B) large negative values of \(x\).
4. Sketch the curve.
5. Solve the inequality \(\frac { ( 2 x - 3 ) ( x + 1 ) } { ( x + 4 ) ( x - 2 ) } \leqslant 2\).

OCR MEI FP1 2006 January Q7

13 marks Standard +0.3

7 A curve has equation \(y = \frac { 3 + x ^ { 2 } } { 4 - x ^ { 2 } }\).

1. Show that \(y\) can never be zero.
2. Write down the equations of the two vertical asymptotes and the one horizontal asymptote.
3. Describe the behaviour of the curve for large positive and large negative values of \(x\), justifying your description.
4. Sketch the curve.
5. Solve the inequality \(\frac { 3 + x ^ { 2 } } { 4 - x ^ { 2 } } \leqslant - 2\).

OCR MEI FP1 2007 January Q7

12 marks Moderate -0.3

7 A curve has equation \(y = \frac { 5 } { ( x + 2 ) ( 4 - x ) }\).

1. Write down the value of \(y\) when \(x = 0\).
2. Write down the equations of the three asymptotes.
3. Sketch the curve.
4. Find the values of \(x\) for which \(\frac { 5 } { ( x + 2 ) ( 4 - x ) } = 1\) and hence solve the inequality $$\frac { 5 } { ( x + 2 ) ( 4 - x ) } < 1 .$$

OCR MEI FP1 2008 January Q7

11 marks Standard +0.3

7 The sketch below shows part of the graph of \(y = \frac { x - 1 } { ( x - 2 ) ( x + 3 ) ( 2 x + 3 ) }\). One section of the graph has been omitted. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{225bff01-f2c4-421f-ac91-c6a0fcb01e6f-3_842_1198_477_552} \captionsetup{labelformat=empty} \caption{Fig. 7}

\end{figure}

1. Find the coordinates of the points where the curve crosses the axes.
2. Write down the equations of the three vertical asymptotes and the one horizontal asymptote.
3. Copy the sketch and draw in the missing section.
4. Solve the inequality \(\frac { x - 1 } { ( x - 2 ) ( x + 3 ) ( 2 x + 3 ) } \geqslant 0\).

OCR MEI FP1 2005 June Q8

14 marks Challenging +1.2

8 A curve has equation \(y = \frac { x ^ { 2 } - 4 } { ( 3 x - 2 ) ^ { 2 } }\).

1. Find the equations of the asymptotes.
2. Describe the behaviour of the curve for large positive and large negative values of \(x\), justifying your description.
3. Sketch the curve.
4. Solve the inequality \(\frac { x ^ { 2 } - 4 } { ( 3 x - 2 ) ^ { 2 } } \geqslant - 1\).

OCR MEI FP1 2008 June Q8

12 marks Standard +0.3

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

1. Write down the equations of the three asymptotes.
2. Determine whether the curve approaches the horizontal asymptote from above or below for
   (A) large positive values of \(x\),
   (B) large negative values of \(x\).
3. Sketch the curve.
4. Solve the inequality \(\frac { 2 x ^ { 2 } } { ( x - 3 ) ( x + 2 ) } < 0\).

Edexcel C1 2014 June Q3

7 marks Moderate -0.8

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

1. \(3 x - 7 > 3 - x\)
2. \(x ^ { 2 } - 9 x \leqslant 36\)
3. both \(3 x - 7 > 3 - x\) and \(x ^ { 2 } - 9 x \leqslant 36\)

Edexcel F2 2018 June Q1

5 marks Moderate -0.5

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

$$\frac { 1 } { x - 2 } > \frac { 2 } { x }$$

OCR C1 2009 January Q8

10 marks Moderate -0.3

8

1. Solve the equation \(5 - 8 x - x ^ { 2 } = 0\), giving your answers in simplified surd form.
2. Solve the inequality \(5 - 8 x - x ^ { 2 } \leqslant 0\).
3. Sketch the curve \(y = \left( 5 - 8 x - x ^ { 2 } \right) ( x + 4 )\), giving the coordinates of the points where the curve crosses the coordinate axes.

OCR C1 2010 January Q11

11 marks Standard +0.3

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 \(x\).

OCR C1 2012 January Q9

12 marks Moderate -0.3

9

1. Sketch the curve \(y = 12 - x - x ^ { 2 }\), giving the coordinates of all intercepts with the axes.
2. Solve the inequality \(12 - x - x ^ { 2 } > 0\).
3. Find the coordinates of the points of intersection of the curve \(y = 12 - x - x ^ { 2 }\) and the line \(3 x + y = 4\).

OCR C1 2009 June Q7

6 marks Moderate -0.8

7

1. Express \(x ^ { 2 } - 5 x + \frac { 1 } { 4 }\) in the form \(( x - a ) ^ { 2 } - b\).
2. Find the centre and radius of the circle with equation \(x ^ { 2 } + y ^ { 2 } - 5 x + \frac { 1 } { 4 } = 0\).

OCR C1 2009 June Q8

6 marks Easy -1.2

8 Solve the inequalities

1. \(- 35 < 6 x + 7 < 1\),
2. \(3 x ^ { 2 } > 48\). \(9 \quad A\) is the point \(( 4 , - 3 )\) and \(B\) is the point \(( - 1,9 )\).

OCR C1 2010 June Q8

10 marks Moderate -0.8

8

1. Express \(2 x ^ { 2 } + 5 x\) in the form \(2 ( x + p ) ^ { 2 } + q\).
2. State the coordinates of the minimum point of the curve \(y = 2 x ^ { 2 } + 5 x\).
3. State the equation of the normal to the curve at its minimum point.
4. Solve the inequality \(2 x ^ { 2 } + 5 x > 0\).

OCR C1 2011 June Q7

8 marks Moderate -0.3

7 Solve the inequalities

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

OCR C1 2012 June Q9

11 marks Moderate -0.3

9

1. A rectangular tile has length \(4 x \mathrm {~cm}\) and width \(( x + 3 ) \mathrm { cm }\). The area of the rectangle is less than \(112 \mathrm {~cm} ^ { 2 }\). By writing down and solving an inequality, determine the set of possible values of \(x\).
2. A second rectangular tile of length \(4 y \mathrm {~cm}\) and width \(( y + 3 ) \mathrm { cm }\) has a rectangle of length \(2 y \mathrm {~cm}\) and width \(y \mathrm {~cm}\) removed from one corner as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{ae6cdd3c-0df9-4fec-b4bd-2237b585c766-3_358_757_479_662} Given that the perimeter of this tile is between 20 cm and 54 cm , determine the set of possible values of \(y\).

OCR C1 2015 June Q8

9 marks Moderate -0.3

8

1. Sketch the curve \(y = 2 x ^ { 2 } - x - 3\), giving the coordinates of all points of intersection with the axes.
2. Hence, or otherwise, solve the inequality \(2 x ^ { 2 } - x - 3 > 0\).
3. Given that the equation \(2 x ^ { 2 } - x - 3 = k\) has no real roots, find the set of possible values of the constant \(k\).