1. 1
   A café charges $x$ pence for a cup of coffee. The café owner wants the price to satisfy the inequality $50 < x \leq 120$.

   (a)

   Which of the following prices satisfies this inequality?

   [1 mark]

   • A45p
   • B50p
   • C90p
   • D125p

   (b)

   Solve the inequality $2x - 8 > 12$ for $x$.

   [1 mark]

   • A$x > 2$
   • B$x > 10$
   • C$x > 4$
   • D$x > 20$

   (c)

   Which set notation correctly describes the solution set of the inequality $x \geq 7$?

   [1 mark]

   • A$\{x : x \geq 7\}$
   • B$\{x : x > 7\}$
   • C$\{x : x \leq 7\}$
   • D$\{x : x < 7\}$

   Total for question 1: 3 marks

2. 2
   A gym membership costs $x$ pounds per month. The gym requires the membership price to satisfy the inequality $4x - 9 \leq 39$ for a promotional discount to apply.

   (a)

   Solve the inequality $4x - 9 \leq 39$ to find the range of values of $x$ for which the promotional discount applies. Show your working.

   [2 marks]

   (b)

   A rival gym offers a different discount if the price satisfies $3(x+2) > 21$. Solve this inequality for $x$, showing your working.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A school is organising a trip. The number of students, $n$, who attend must satisfy $2n + 15 \leq 75$ due to bus capacity, and at least 20 students must attend for the trip to go ahead, so $n \geq 20$.

   (a)

   Solve the inequality $2n + 15 \leq 75$ to find the maximum number of students, $n$, who can attend, and combine your result with the requirement that at least 20 students must attend ($n \geq 20$) to give the full range of values of $n$. Show your method.

   [3 marks]

   (b)

   Show that the combined solution set $20 \leq n \leq 30$ contains exactly 11 possible whole-number values of $n$. Give full reasoning.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A manufacturer produces rectangular metal plates. The area of a plate, in cm², is given by $A = x^2 + 2x$, where $x$ is the plate's width in cm. Quality control requires the width to satisfy $x^2 + 2x - 24 \leq 0$, and a valid width must also satisfy $x > 0$.

   (a)

   Solve the quadratic inequality $x^2 + 2x - 24 \leq 0$, giving your answer using set notation. Show your working.

   [4 marks]

   (b)

   Given also that a valid width must satisfy $x > 0$, find the range of values of $x$ that satisfy both inequalities. Explain your reasoning.

   [4 marks]

   (c)

   Using the range of valid widths found in the previous part, calculate how many whole-number widths (in cm) are valid, and calculate the maximum possible area for a valid plate. Show all your working.

   [5 marks]

   Total for question 4: 13 marks

End of questions