Algebraic and modulus inequalitiesEdexcel International A Level Further Maths: Subtopic test

1. 1
   Consider the inequality $\frac{x}{x-3}>2$.

   (a)

   Which of these are the critical values for this inequality?

   [1 mark]

   • A$x=3$ and $x=6$
   • B$x=0$ and $x=3$
   • C$x=2$ and $x=3$
   • D$x=6$ only

   (b)

   Which of these is the solution of the inequality?

   [1 mark]

   • A$x<3$ or $x>6$
   • B$x<6$
   • C$x>6$
   • D$3<x<6$

   (c)

   Hence solve $\frac{x}{x-3}\le2$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the inequality $|2x-3|<x+2$.

   (a)

   Which of these are the critical values, found by solving $|2x-3|=x+2$?

   [1 mark]

   • A$x=5$ only
   • B$x=\frac13$ and $x=5$
   • C$x=-\frac53$ and $x=5$
   • D$x=-2$ and $x=5$

   (b)

   Which of these is the solution of the inequality $|2x-3|<x+2$?

   [1 mark]

   • A$x<\frac13$ or $x>5$
   • B$x>\frac13$
   • C$\frac13<x<5$
   • D$-2<x<5$

   (c)

   Hence write down all the integers $x$ that satisfy $|2x-3|<x+2$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the inequality $\frac{1}{x-1}>\frac{x}{x-4}$.

   (a)

   Show that the inequality is equivalent to $(x-1)(x-4)\left(x^2-2x+4\right)<0$.

   [3 marks]

   (b)

   Hence solve $\frac{1}{x-1}>\frac{x}{x-4}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   In each part, solve the inequality, giving the exact critical values and showing your reasoning.

   (a)

   Solve $|x^2-1|>2(x+1)$.

   [6 marks]

   (b)

   Solve $\frac{1}{x-5}>\frac{x}{x-8}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions