1.02g Inequalities: linear and quadratic in single variable

420 questions

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OCR C1 2014 June Q5
8 marks Moderate -0.3
Solve the following inequalities.
  1. \(5 < 6x + 3 < 14\) [3]
  2. \(x(3x - 13) \geqslant 10\) [5]
OCR MEI C1 Q1
3 marks Easy -1.2
Solve the inequality \(2(x - 3) < 6x + 15\). [3]
OCR MEI C1 Q12
12 marks Moderate -0.3
  1. Show that the graph of \(y = x^2 - 3x + 11\) is above the \(x\)-axis for all values of \(x\). [3]
  2. Find the set of values of \(x\) for which the graph of \(y = 2x^2 + x - 10\) is above the \(x\)-axis. [4]
  3. Find algebraically the coordinates of the points of intersection of the graphs of $$y = x^2 - 3x + 11 \quad\text{and}\quad y = 2x^2 + x - 10.$$ [5]
OCR MEI C1 2006 January Q4
4 marks Easy -1.8
Solve the inequality \(\frac{3(2x + 1)}{4} > -6\). [4]
OCR MEI C1 2006 June Q6
4 marks Moderate -0.8
Solve the inequality \(x^2 + 2x < 3\). [4]
OCR MEI C1 2009 June Q4
2 marks Easy -1.2
Solve the inequality \(x(x - 6) > 0\). [2]
OCR MEI C1 2010 June Q4
5 marks Easy -1.2
Solve the following inequalities.
  1. \(2(1 - x) > 6x + 5\) [3]
  2. \((2x - 1)(x + 4) < 0\) [2]
OCR MEI C1 2011 June Q1
3 marks Easy -1.8
Solve the inequality \(6(x + 3) > 2x + 5\). [3]
Edexcel C1 Q4
6 marks Moderate -0.8
  1. Solve the inequality $$x^2 + 3x > 10.$$ [3]
  2. Find the set of values of \(x\) which satisfy both of the following inequalities: $$3x - 2 < x + 3$$ $$x^2 + 3x > 10$$ [3]
Edexcel C1 Q4
6 marks Moderate -0.8
  1. Find the value of the constant \(k\) such that the equation $$x^2 - 6x + k = 0$$ has equal roots. [2]
  2. Solve the inequality $$2x^2 - 9x + 4 < 0.$$ [4]
Edexcel C1 Q6
8 marks Moderate -0.8
  1. Sketch on the same diagram the curve with equation \(y = (x - 2)^2\) and the straight line with equation \(y = 2x - 1\). Label on your sketch the coordinates of any points where each graph meets the coordinate axes. [5]
  2. Find the set of values of \(x\) for which $$(x - 2)^2 > 2x - 1.$$ [3]
Edexcel C1 Q4
6 marks Moderate -0.8
Find the set of values of \(x\) for which
  1. \(6x - 11 > x + 4\), [2]
  2. \(x^2 - 6x - 16 < 0\), [3]
  3. both \(6x - 11 > x + 4\) and \(x^2 - 6x - 16 < 0\). [1]
OCR C1 Q2
4 marks Moderate -0.3
Find the set of values of \(x\) for which $$(x - 1)(x - 2) < 20.$$ [4]
OCR C1 Q3
5 marks Moderate -0.8
Find the set of values of \(x\) for which
  1. \(6x - 11 > x + 4\), [2]
  2. \(x^2 - 6x - 16 < 0\). [3]
OCR MEI C1 Q5
12 marks Moderate -0.3
  1. Write \(x^2 - 5x + 8\) in the form \((x - a)^2 + b\) and hence show that \(x^2 - 5x + 8 > 0\) for all values of \(x\). [4]
  2. Sketch the graph of \(y = x^2 - 5x + 8\), showing the coordinates of the turning point. [3]
  3. Find the set of values of \(x\) for which \(x^2 - 5x + 8 > 14\). [3]
  4. If \(f(x) = x^2 - 5x + 8\), does the graph of \(y = f(x) - 10\) cross the \(x\)-axis? Show how you decide. [2]
OCR C3 2013 January Q3
7 marks Standard +0.8
  1. Given that \(|t| = 3\), find the possible values of \(|2t - 1|\). [3]
  2. Solve the inequality \(|x - t^2| > |x + 3\sqrt{2}|\). [4]
OCR C3 2010 June Q5
7 marks Standard +0.8
  1. Solve the inequality \(|2x + 1| \leqslant |x - 3|\). [5]
  2. Given that \(x\) satisfies the inequality \(|2x + 1| \leqslant |x - 3|\), find the greatest possible value of \(|x + 2|\). [2]
OCR C3 2010 June Q9
13 marks Standard +0.3
The functions f and g are defined for all real values of \(x\) by $$f(x) = 4x^2 - 12x \quad \text{and} \quad g(x) = ax + b,$$ where \(a\) and \(b\) are non-zero constants.
  1. Find the range of f. [3]
  2. Explain why the function f has no inverse. [2]
  3. Given that \(g^{-1}(x) = g(x)\) for all values of \(x\), show that \(a = -1\). [4]
  4. Given further that gf\((x) < 5\) for all values of \(x\), find the set of possible values of \(b\). [4]
OCR C3 Q2
5 marks Standard +0.8
Find the set of values of \(x\) such that $$|3x + 1| \leq |x - 2|.$$ [5]
AQA FP1 2014 June Q6
10 marks Standard +0.3
A curve \(C\) has equation \(y = \frac{1}{x(x + 2)}\).
  1. Write down the equations of all the asymptotes of \(C\). [2 marks]
  2. The curve \(C\) has exactly one stationary point. The \(x\)-coordinate of the stationary point is \(-1\).
    1. Find the \(y\)-coordinate of the stationary point. [1 mark]
    2. Sketch the curve \(C\). [2 marks]
  3. Solve the inequality $$\frac{1}{x(x + 2)} \leqslant \frac{1}{8}$$ [5 marks]
AQA FP1 2016 June Q9
11 marks Standard +0.3
A curve \(C\) has equation \(y = \frac{x - 1}{(x - 2)(2x - 1)}\). The line \(L\) has equation \(y = \frac{1}{2}(x - 1)\).
  1. Write down the equations of the asymptotes of \(C\). [2 marks]
  2. By forming and solving a suitable cubic equation, find the \(x\)-coordinates of the points of intersection of \(L\) and \(C\). [3 marks]
  3. Given that \(C\) has no stationary points, sketch \(C\) and \(L\) on the same axes. [3 marks]
  4. Hence solve the inequality \(\frac{x - 1}{(x - 2)(2x - 1)} \geqslant \frac{1}{2}(x - 1)\). [3 marks]
OCR MEI FP1 2006 June Q7
13 marks Standard +0.3
A curve has equation \(y = \frac{x^2}{(x-2)(x+1)}\).
  1. Write down the equations of the three asymptotes. [3]
  2. Determine whether the curve approaches the horizontal asymptote from above or from below for
    1. large positive values of \(x\),
    2. large negative values of \(x\). [3]
  3. Sketch the curve. [4]
  4. Solve the inequality \(\frac{x^2}{(x-2)(x+1)} > 0\). [3]
OCR H240/03 2019 June Q1
2 marks Easy -1.2
\includegraphics{figure_1} The diagram shows triangle \(ABC\), with \(AC = 13.5\) cm, \(BC = 8.3\) cm and angle \(ABC = 32°\). Find angle \(CAB\). [2]
OCR H240/03 2020 November Q3
11 marks Moderate -0.8
The functions f and g are defined for all real values of x by \(f(x) = 2x^2 + 6x\) and \(g(x) = 3x + 2\).
  1. Find the range of f. [3]
  2. Give a reason why f has no inverse. [1]
  3. Given that \(fg(-2) = g^{-1}(a)\), where \(a\) is a constant, determine the value of \(a\). [4]
  4. Determine the set of values of \(x\) for which \(f(x) > g(x)\). Give your answer in set notation. [3]
AQA AS Paper 1 2019 June Q5
5 marks Moderate -0.8
  1. Sketch the curve \(y = g(x)\) where $$g(x) = (x + 2)(x - 1)^2$$ [3 marks]
  2. Hence, solve \(g(x) \leq 0\) [2 marks]