Further Pure 1: InequalitiesEdexcel A-Level Further Maths: Topic test

1. 1
   The inequality $|3x+1|\geqslant7$ is to be solved.

   (a)

   Which pair of conditions is equivalent to $|3x+1|\geqslant7$?

   [1 mark]

   • A$3x+1\geqslant7$ and $3x+1\leqslant-7$
   • B$3x+1\geqslant7$ or $3x+1\geqslant-7$
   • C$3x+1\geqslant7$ or $3x+1\leqslant-7$
   • D$-7\leqslant3x+1\leqslant7$

   (b)

   What is the solution of $|3x+1|\geqslant7$?

   [1 mark]

   • A$x\leqslant-\frac83$ or $x\geqslant2$
   • B$-\frac83\leqslant x\leqslant2$
   • C$x\leqslant-\frac73$ or $x\geqslant2$
   • D$x\leqslant-2$ or $x\geqslant2$

   (c)

   Find the values of $x$ for which both $|3x+1|\geqslant7$ and $x^2<9$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The inequality $\frac{x}{x-4}<3$ is to be solved, where $x\neq4$.

   (a)

   Multiplying both sides of the inequality by $(x-4)^2$ gives which inequality?

   [1 mark]

   • A$x<3(x-4)$
   • B$x(x-4)>3(x-4)^2$
   • C$x^2<3(x-4)^2$
   • D$x(x-4)<3(x-4)^2$

   (b)

   What is the solution of $\frac{x}{x-4}<3$?

   [1 mark]

   • A$x<6$
   • B$4<x<6$
   • C$x>6$
   • D$x<4$ or $x>6$

   (c)

   Find the values of $x$ for which $\frac{x}{x-4}<3$ and $x>0$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The inequality $|x^2-6|>3x$ is to be solved.

   (a)

   Find the critical values of $x$ by solving $x^2-6=3x$ and $x^2-6=-3x$.

   [3 marks]

   (b)

   Hence solve $|x^2-6|>3x$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $\mathrm{f}(x)=\frac{2}{x-1}$ and $\mathrm{g}(x)=\frac{x}{x+2}$, where $x\neq1$ and $x\neq-2$.

   (a)

   Solve the inequality $\mathrm{f}(x)>\mathrm{g}(x)$.

   [6 marks]

   (b)

   Hence solve $\frac{2}{|x|-1}>\frac{|x|}{|x|+2}$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Let $\mathrm{h}(x)=\frac{(x+3)(x-1)}{x-2}$, where $x\neq2$.

   (a)

   Which of these values of $x$ satisfies $\mathrm{h}(x)\geqslant0$?

   [1 mark]

   • A$x=-4$
   • B$x=1.5$
   • C$x=2$
   • D$x=-1$

   (b)

   What is the solution of $\mathrm{h}(x)\geqslant0$?

   [1 mark]

   • A$-3<x<1$ or $x>2$
   • B$-3\leqslant x\leqslant1$ or $x>2$
   • C$-3\leqslant x\leqslant1$ or $x\geqslant2$
   • D$x\leqslant-3$ or $1\leqslant x<2$

   (c)

   Find the values of $x$ for which $\mathrm{h}(x)\geqslant0$ and $x^2<4$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The inequality $|x+2|>|2x-1|$ is to be solved.

   (a)

   Which inequality is obtained by squaring both sides?

   [1 mark]

   • A$(x+2)^2>(2x-1)^2$
   • B$x+2>2x-1$
   • C$(x+2)^2>2x-1$
   • D$x^2+4>4x^2+1$

   (b)

   What is the solution of $|x+2|>|2x-1|$?

   [1 mark]

   • A$x<-\frac13$ or $x>3$
   • B$-3<x<\frac13$
   • C$-\frac13<x<3$
   • D$x<3$

   (c)

   Hence find the values of $x$ for which $|x+2|\leqslant|2x-1|$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The inequality $\frac{3x}{x+1}\geqslant\frac{x}{x-2}$ is to be solved, where $x\neq-1$ and $x\neq2$.

   (a)

   Show that the inequality is equivalent to $\frac{x(2x-7)}{(x+1)(x-2)}\geqslant0$.

   [3 marks]

   (b)

   Hence solve the inequality.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Let $\mathrm{f}(x)=|x^2-9|$ and $\mathrm{g}(x)=2(x+3)$.

   (a)

   Solve the inequality $\mathrm{f}(x)>\mathrm{g}(x)$.

   [6 marks]

   (b)

   Find the set of values of $x$ for which both $\mathrm{f}(x)>\mathrm{g}(x)$ and $\frac{x-2}{x+1}<1$.

   [6 marks]

   Total for question 8: 12 marks

End of questions