Linear and quadratic inequalitiesEdexcel International A Level Maths: Subtopic test

1. 1
   Consider the inequality $3x-7>5x+1$, where $x$ is a real number.

   (a)

   Which of the following is the solution of $3x-7>5x+1$?

   [1 mark]

   • A$x>-4$
   • B$x<-4$
   • C$x>4$
   • D$x<4$

   (b)

   Which of the following describes the set of values of $x$ satisfying both $3x-7>5x+1$ and $x>-10$?

   [1 mark]

   • A$x<-4$
   • B$x>-10$
   • C$x<-10$ or $x>-4$
   • D$-10<x<-4$

   (c)

   Find the set of values of $x$ for which $3x-7>5x+1$ or $2x+6>0$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The function $f(x)=x^2-2x-8$ is defined for real $x$.

   (a)

   Which of the following is the set of values of $x$ for which $f(x)<0$?

   [1 mark]

   • A$-2<x<4$
   • B$x<-2$ or $x>4$
   • C$-4<x<2$
   • D$x<-4$ or $x>2$

   (b)

   Which of the following is the set of values of $x$ for which $f(x)>7$?

   [1 mark]

   • A$-3<x<5$
   • B$x>5$
   • C$x<-3$ or $x>5$
   • D$x<-5$ or $x>3$

   (c)

   Find the set of values of $x$ for which $f(x)<0$ and $2x-1>0$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A rectangular patio has width $x$ m and length $(x+6)$ m. Its area must be less than $55$ m$^2$, and its width must be greater than $2$ m.

   (a)

   Show that $x^2+6x-55<0$ and hence find the range of possible values of $x$.

   [3 marks]

   (b)

   The perimeter of the patio must be at least $28$ m. Find the range of values of $x$ satisfying all three conditions.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The curve $C_1$ has equation $y=x^2-4x+7$, the line $L$ has equation $y=x+1$ and the curve $C_2$ has equation $y=\frac{6}{x}$ for $x\neq0$, where $x$ is real.

   (a)

   (i) Find the set of values of $x$ for which $C_1$ lies below $L$.
   (ii) Find the set of values of $x$ for which $C_1$ lies above the line $y=3$.
   (iii) Hence find the set of values of $x$ for which $C_1$ lies below $L$ and above $y=3$.

   [6 marks]

   (b)

   Find the set of values of $x$ for which $C_2$ lies below $L$.

   [6 marks]

   Total for question 4: 12 marks

End of questions