FP2: InequalitiesEdexcel International A Level Further Maths: Topic test

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

   (a)

   Find the critical values of $x$.

   [1 mark]

   • A$-2$ and $5$
   • B$1$ only
   • C$-2$ and $1$
   • D$-5$ and $-2$

   (b)

   Which of the following is the solution set of the inequality?

   [1 mark]

   • A$x<-2$ or $x>1$
   • B$x>1$
   • C$-2<x<1$
   • D$x<-2$

   (c)

   A student multiplies both sides of the inequality by $x+2$ and concludes that $x>1$. Explain why this method is not valid and give a value of $x$ that satisfies the original inequality but not the student's answer.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Consider the inequality $|x+1|\ge5$ and the equation $|x+1|=2x-1$.

   (a)

   Solve the inequality $|x+1|\ge5$.

   [1 mark]

   • A$-6\le x\le4$
   • B$x\le-4$ or $x\ge6$
   • C$x\le-5$ or $x\ge5$
   • D$x\le-6$ or $x\ge4$

   (b)

   How many integers $x$ do not satisfy $|x+1|\ge5$?

   [1 mark]

   • A$8$
   • B$9$
   • C$10$
   • D$11$

   (c)

   Solve the equation $|x+1|=2x-1$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Consider the inequality $\frac{2}{x-1}\le\frac{1}{x+2}$, where $x\neq1$ and $x\neq-2$.

   (a)

   Show that the inequality is equivalent to $\frac{x+5}{(x-1)(x+2)}\le0$.

   [3 marks]

   (b)

   Hence solve the inequality.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Let $g(x)=x^2-5x$ for real $x$.

   (a)

   Solve the inequality $|g(x)|<6$.

   [6 marks]

   (b)

   Solve the inequality $\frac{6}{x}>x-5$, where $x\neq0$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Consider the inequality $\frac{x^2}{x+3}<4$, where $x\neq-3$.

   (a)

   Which inequality is equivalent to the one given?

   [1 mark]

   • A$\frac{(x+6)(x-2)}{x+3}<0$
   • B$\frac{(x-6)(x+2)}{x+3}<0$
   • C$(x-6)(x+2)<0$
   • D$\frac{(x-2)(x+2)}{x+3}<0$

   (b)

   Which of the following is the solution set of the inequality?

   [1 mark]

   • A$-2<x<6$
   • B$-3<x<-2$ or $x>6$
   • C$x<-3$ or $-2<x<6$
   • D$x<-3$ or $x>6$

   (c)

   Find the sum of all the positive integers that satisfy the inequality.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Consider the inequality $|3x-1|\le|x+5|$.

   (a)

   Squaring both sides of the inequality gives which quadratic inequality, in its simplest form?

   [1 mark]

   • A$x^2-2x-3\le0$
   • B$x^2-2x-3\ge0$
   • C$2x^2+x-6\le0$
   • D$x^2+2x-3\le0$

   (b)

   Which of the following is the solution set of the inequality?

   [1 mark]

   • A$x\le-1$ or $x\ge3$
   • B$-3\le x\le1$
   • C$-1<x<3$
   • D$-1\le x\le3$

   (c)

   Explain why squaring both sides of $|3x-1|\le|x+5|$ gives an equivalent inequality for all real $x$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Consider the inequality $\frac{x}{x-1}>\frac{x+2}{x+4}$, where $x\neq1$ and $x\neq-4$.

   (a)

   Show that the critical values of $x$ are $-4$, $-\frac23$ and $1$.

   [3 marks]

   (b)

   Hence solve the inequality.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Let $f(x)=|x^2-9|$ and $h(x)=2(x+3)$ for real $x$.

   (a)

   Solve the inequality $f(x)>h(x)$.

   [6 marks]

   (b)

   Hence, or otherwise, solve $\frac{f(x)}{x+3}>2$.

   [6 marks]

   Total for question 8: 12 marks

End of questions