Linear and quadratic inequalitiesEdexcel A-Level Maths: Subtopic test

1. 1
   The inequality $7-2x<3x+1$ is given.

   (a)

   Solve the inequality.

   [1 mark]

   • A$x<\frac65$
   • B$x>\frac65$
   • C$x>\frac85$
   • D$x>-\frac65$

   (b)

   How many integers $x$ satisfy both $7-2x<3x+1$ and $x<6$?

   [1 mark]

   • A3
   • B5
   • C4
   • D6

   (c)

   Find the set of values of $x$ that satisfy both $7-2x<3x+1$ and $x^2\le16$. Give your answer in set notation.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A curve has equation $y=x^2-3x-10$.

   (a)

   Which set of values of $x$ satisfies $x^2-3x-10<0$?

   [1 mark]

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

   (b)

   Which statement describes the values of $x$ for which the curve lies on or above the $x$-axis?

   [1 mark]

   • A$x\le-2$ or $x\ge5$
   • B$-2\le x\le5$
   • C$x\le-2$ and $x\ge5$
   • D$x\le5$ or $x\ge-2$

   (c)

   Find the set of values of $x$ for which the curve lies below the line $y=2x+4$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A student is solving inequalities involving the expression $\frac{6}{x+2}$, where $x\neq-2$.

   (a)

   Solve $\frac{6}{x+2}<3$.

   [3 marks]

   (b)

   Find the set of values of $x$ that satisfy both $\frac{6}{x+2}<3$ and $x^2<9$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A rectangular pen is built against a straight wall. It uses exactly $40$ m of fencing for the three sides that are not on the wall. The two sides perpendicular to the wall each have length $x$ m. Fencing costs £5 per metre for the two perpendicular sides and £8 per metre for the side parallel to the wall.

   (a)

   (i) Show that the area of the pen is $x(40-2x)$ m$^2$, and that an area of at least $150$ m$^2$ requires $x^2-20x+75\le0$.
   (ii) Solve this inequality.
   (iii) The side parallel to the wall must be at least $12$ m long. Find the range of possible values of $x$.

   [6 marks]

   (b)

   (i) Show that the total cost of the fencing is £$(320-6x)$.
   (ii) The farmer has a budget of £$260$. Find the values of $x$ for which the cost does not exceed the budget.
   (iii) Hence find all values of $x$ that satisfy the budget and the conditions in part (a).

   [6 marks]

   Total for question 4: 12 marks

End of questions