Algebraic inequalitiesEdexcel A-Level Further Maths: Subtopic test

1. 1
   The inequality $\frac{x-1}{x+2}<2$ is to be solved, where $x\neq-2$.

   (a)

   A student begins by multiplying both sides by $(x+2)$. Why is this not valid as it stands?

   [1 mark]

   • AInequalities can never be multiplied by an expression containing $x$.
   • B$(x+2)$ can be negative, which would reverse the inequality, and its sign is not known.
   • C$(x+2)$ is always positive, so the inequality is unchanged but nothing is gained.
   • DThe inequality must first be squared.

   (b)

   Which of these is the solution set?

   [1 mark]

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

   (c)

   The student's method gives $x-1<2(x+2)$, which leads to $x>-5$. Show that $x=-3$ satisfies $x>-5$ but is not a solution of the original inequality, and explain the error.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The inequality $|2x-3|<x+4$ is to be solved.

   (a)

   Which statement must be true for every $x$ satisfying the inequality?

   [1 mark]

   • A$x+4>0$
   • B$x>7$
   • C$2x-3<0$
   • D$x<-4$

   (b)

   Which of these is the solution set?

   [1 mark]

   • A$-4<x<7$
   • B$x<-\frac13$ or $x>7$
   • C$-\frac13<x<7$
   • D$-4<x<-\frac13$

   (c)

   Hence solve $|2x-3|\geq x+4$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the inequality $\frac{3}{x-2}>\frac{2}{x+1}$, where $x\neq2$ and $x\neq-1$.

   (a)

   Show that the inequality is equivalent to $(x-2)(x+1)(x+7)>0$.

   [3 marks]

   (b)

   Hence solve the inequality.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $f(x)=|x^{2}-4|$. Give each solution as a range or ranges of $x$.

   (a)

   Solve $f(x)>3x$.

   [6 marks]

   (b)

   Solve $f(x)<x+2$.

   [6 marks]

   Total for question 4: 12 marks

End of questions