1.02g Inequalities: linear and quadratic in single variable

420 questions

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OCR MEI C1 Q2
3 marks Easy -1.2
2 Find the range of values of \(x\) for which \(x ^ { 2 } - 5 x + 6 \leq 0\).
OCR MEI C1 Q6
3 marks Easy -1.8
6 List the integers which satisfy both of the following inequalities: $$2 x - 9 < 0 , \quad 8 - x \leq 6$$
OCR MEI C1 Q10
12 marks Moderate -0.8
10
  1. A quadratic function is given by \(\mathrm { f } ( x ) = x ^ { 2 } - 6 x + 8\).
    Sketch the graph of \(y = \mathrm { f } ( x )\), giving the coordinates of the points where it crosses the axes. Mark the lowest point on the curve, and give its coordinates.
  2. Solve the inequality \(x ^ { 2 } - 6 x + 8 < 0\).
  3. On the same graph, sketch \(y = \mathrm { f } ( x + 3 )\).
  4. The graph of \(y = \mathrm { f } ( x + 3 ) - 2\) is obtained from the graph of \(y = \mathrm { f } ( x )\) by a transformation. Describe the transformation and sketch the curve on the same axes as in (i) and (iii) above. Label all these curves clearly.
OCR MEI C1 Q3
4 marks Moderate -0.5
3 Solve the inequality \(2 x ^ { 2 } - 7 x \geq 4\).
OCR MEI C1 Q6
3 marks Easy -1.2
6 Find the positive integer values of \(x\) for which $$\frac { 1 } { 2 } ( 26 - 2 x ) \geq 2 ( 3 + x )$$
OCR C1 Q4
4 marks Moderate -0.5
4. Solve the inequality $$2 x ^ { 2 } - 9 x + 4 < 0 .$$
OCR C1 Q5
6 marks Moderate -0.3
  1. Solve the inequality $$x ^ { 2 } + 3 x > 10 .$$
  2. Find the set of values of \(x\) which satisfy both of the following inequalities: $$\begin{aligned} & 3 x - 2 < x + 3 \\ & x ^ { 2 } + 3 x > 10 \end{aligned}$$
OCR C1 Q5
7 marks Moderate -0.8
  1. Sketch on the same diagram the curve with equation \(y = ( x - 2 ) ^ { 2 }\) and the straight line with equation \(y = 2 x - 1\). Label on your sketch the coordinates of any points where each graph meets the coordinate axes.
  2. Find the set of values of \(x\) for which $$( x - 2 ) ^ { 2 } > 2 x - 1$$
OCR C1 Q3
5 marks Moderate -0.3
3. $$f ( x ) = x ^ { 3 } + 4 x ^ { 2 } - 3 x + 7$$ Find the set of values of \(x\) for which \(\mathrm { f } ( x )\) is increasing.
OCR C1 Q5
6 marks Moderate -0.8
5. Given that the equation $$x ^ { 2 } + 4 k x - k = 0$$ has no real roots,
  1. show that $$4 k ^ { 2 } + k < 0 ,$$
  2. find the set of possible values of \(k\).
OCR C1 Q1
4 marks Moderate -0.3
  1. Solve the inequality
$$x ( 2 x + 1 ) \leq 6 .$$
OCR C1 Q1
3 marks Easy -1.8
  1. Solve the inequality
$$4 ( x - 2 ) < 2 x + 5$$
OCR MEI C1 Q1
3 marks Easy -1.8
1 Solve the inequality \(\frac { 4 x - 5 } { 7 } > 2 x + 1\).
OCR MEI C1 Q2
3 marks Moderate -0.8
2 Solve the inequality \(3 x ^ { 2 } + 10 x + 3 > 0\).
OCR MEI C1 Q3
4 marks Moderate -0.8
3 Solve the inequality \(5 x ^ { 2 } - 28 x - 12 \leqslant 0\).
OCR MEI C1 Q4
4 marks Easy -1.8
4 Solve the following inequality. $$\frac { 2 x + 1 } { 5 } < \frac { 3 x + 4 } { 6 }$$
OCR MEI C1 Q5
3 marks Easy -1.8
5 Solve the inequality \(6 ( x + 3 ) > 2 x + 5\).
OCR MEI C1 Q6
2 marks Easy -1.8
6 Solve the inequality \(5 - 2 x < 0\).
OCR MEI C1 Q7
5 marks Easy -1.2
7 Solve the following inequalities.
  1. \(2 ( 1 - x ) > 6 x + 5\)
  2. \(( 2 x - 1 ) ( x + 4 ) < 0\)
OCR MEI C1 Q8
3 marks Easy -1.8
8 Solve the inequality \(\frac { 5 x - 3 } { 2 } < x + 5\).
OCR MEI C1 Q9
2 marks Easy -1.2
9 Solve the inequality \(x ( x - 6 ) > 0\).
OCR MEI C1 Q10
3 marks Easy -1.8
10 Solve the inequality \(7 - x < 5 x - 2\).
OCR MEI C1 Q11
2 marks Easy -1.8
11 Solve the inequality \(3 x - 1 > 5 - x\).
OCR MEI C1 Q12
3 marks Easy -1.8
12 Solve the inequality \(1 - 2 x < 4 + 3 x\).
OCR MEI C1 Q13
4 marks Easy -1.2
13 Solve the inequality \(x ^ { 2 } + 2 x < 3\).