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.

35 questions · Standard +0.2

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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 Q5
7 marks Standard +0.3
5. Using algebra, find the set of values of \(x\) for which $$2 x - 5 > \frac { 3 } { x }$$ [P4 June 2002 Qn 4]
Edexcel FP2 Q18
6 marks Standard +0.3
18. Solve the inequality \(\frac { 1 } { 2 x + 1 } > \frac { x } { 3 x - 2 }\).
AQA Further AS Paper 1 2020 June Q14
7 marks Standard +0.3
14
  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.
    14
  2. Briefly explain why this statement is incorrect. $$\frac { ( x + p ) ( x + q ) } { x + r } \geq 0 \Leftrightarrow ( x + p ) ( x + q ) ( x + r ) \geq 0$$ 14
  3. Solve $$\frac { x + 7 } { x + 1 } \leq x + 1$$
AQA Further Paper 2 2020 June Q5
5 marks Moderate -0.5
5 Solve the inequality $$\frac { 2 x + 3 } { x - 1 } \leq x + 5$$