Rational inequality algebraically

Solve inequalities of the form f(x)/g(x) > h(x) or similar by algebraic manipulation, finding critical points, and sign analysis.

40 questions · Standard +0.2

1.02g Inequalities: linear and quadratic in single variable
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Edexcel FP1 AS 2020 June Q2
5 marks Standard +0.8
  1. Use algebra to determine the values of \(x\) for which
$$\frac { x + 1 } { 2 x ^ { 2 } + 5 x - 3 } > \frac { x } { 4 x ^ { 2 } - 1 }$$
Edexcel FP1 AS 2021 June Q1
6 marks Standard +0.3
  1. Use algebra to determine the values of \(x\) for which
$$x ( x - 1 ) > \frac { x - 1 } { x }$$ giving your answer in set notation.
Edexcel FP1 AS 2022 June Q1
6 marks Standard +0.3
  1. Use algebra to find the set of values of \(x\) for which
$$x \geqslant \frac { 2 x + 15 } { 2 x + 3 }$$
Edexcel FP1 AS 2023 June Q1
5 marks Moderate -0.3
  1. (a) Use algebra to determine the values of \(x\) for which
$$\frac { 5 x } { x - 2 } \geqslant 12$$ (b) Hence, given that \(x\) is an integer, deduce the value of \(x\).
Edexcel FP1 AS 2024 June Q1
7 marks Standard +0.3
  1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.
    1. Sketch the graph of the curve with equation
    $$y = \frac { 1 } { x ^ { 2 } }$$
  2. Solve, using algebra, the inequality $$3 - 2 x ^ { 2 } > \frac { 1 } { x ^ { 2 } }$$
Edexcel FP1 AS Specimen Q3
6 marks Standard +0.3
  1. Use algebra to find the set of values of x for which
$$\frac { 1 } { x } < \frac { x } { x + 2 }$$
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Edexcel FP2 Q3
7 marks Standard +0.8
  1. Find the set of values of \(x\) for which $$x + 4 > \frac{2}{x+3}$$ [6]
  2. Deduce, or otherwise find, the values of \(x\) for which $$x + 4 > \left|\frac{2}{x+3}\right|$$ [1]
Edexcel FP2 Q1
7 marks Standard +0.3
Find the set of values of \(x\) for which $$\frac{3}{x+3} > \frac{x-4}{x}.$$ [7]
Edexcel FP2 Q2
7 marks Standard +0.3
Use algebra to find the set of values of \(x\) for which $$\frac{6x}{3 - x} > \frac{x + 1}{1}$$ [7]
Edexcel FP2 2008 June Q2
Standard +0.3
  1. Simplify the expression \(\frac{(x + 3)(x + 9)}{x - 1} - (3x - 5)\), giving your answer in the form \(\frac{a(x + b)(x + c)}{x - 1}\), where \(a\), \(b\) and \(c\) are integers. (4)
  2. Hence, or otherwise, solve the inequality \(\frac{(x + 3)(x + 9)}{x - 1} > 3x - 5\) (4)(Total 8 marks)
Edexcel FP2 Q5
7 marks Standard +0.3
Using algebra, find the set of values of \(x\) for which $$2x - 5 > \frac{3}{x}.$$ [7]
Edexcel FP2 Q18
6 marks Standard +0.8
Solve the inequality \(\frac{1}{2x + 1} > \frac{x}{3x - 2}\). [6]
AQA AS Paper 2 2023 June Q9
6 marks Easy -1.2
A craft artist is producing items and selling them in a local market. The selling price, £P, of an item is inversely proportional to the number of items produced, \(n\)
  1. When \(n = 10\), \(P = 24\) Show that \(P = \frac{240}{n}\) [1 mark]
  2. The production cost, £C, of an item is inversely proportional to the square of the number of items produced, \(n\) When \(n = 10\), \(C = 60\) Find the set of values of \(n\) for which \(P > C\) [4 marks]
  3. Explain the significance to the craft artist of the range of values found in part (b). [1 mark]
AQA Further AS Paper 1 2020 June Q14
7 marks Standard +0.8
  1. Given $$\frac{x + 7}{x + 1} \leq x + 1$$ show that $$\frac{(x + a)(x + b)}{x + c} \geq 0$$ where \(a\), \(b\), and \(c\) are integers to be found. [4 marks]
  2. Briefly explain why this statement is incorrect. $$\frac{(x + p)(x + q)}{x + r} \geq 0 \Leftrightarrow (x + p)(x + q)(x + r) \geq 0$$ [1 mark]
  3. Solve $$\frac{x + 7}{x + 1} \leq x + 1$$ [2 marks]
AQA Further Paper 2 2020 June Q5
5 marks Standard +0.3
Solve the inequality $$\frac{2x + 3}{x - 1} \leq x + 5$$ [5 marks]