1. 1
   A lift has a weight limit represented by the inequality $w \leq 800$, where $w$ is the total weight in kg carried.

   (a)

   A lift has a weight limit represented by $w \leq 800$, where $w$ is the total weight in kg. If the lift currently holds $650$ kg, how much more weight, $x$ kg, can it safely take, expressed as an inequality?

   [1 mark]

   • A$x \leq 800$
   • B$x \leq 150$
   • C$x < 150$
   • D$x \geq 150$

   (b)

   On a number line, which type of circle is used to represent the boundary value in the inequality $w \leq 800$?

   [1 mark]

   • ANo circle is drawn
   • BOpen circle
   • CClosed (filled) circle
   • DA square marker

   (c)

   A second lift has a strict limit of $w < 500$. Which statement is true of $w = 500$ kg?

   [1 mark]

   • AIt cannot be determined
   • BIt is permitted
   • CIt is the minimum allowed weight
   • DIt is not permitted

   Total for question 1: 3 marks

2. 2
   A delivery van can carry a maximum load related to the number of parcels $x$ it holds, modelled by the inequality $3x < 2x + 4$.

   (a)

   Solve the inequality $3x < 2x + 4$, and represent the solution on a number line, stating whether the boundary point uses an open or closed circle.

   [2 marks]

   (b)

   Solve the compound inequality $-3 \leq 3x - 2 < 7$, giving your answer as a single inequality in $x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two gyms offer different membership pricing. A student is comparing costs over $m$ months to decide which gym is better value.

   (a)

   A gym charges a joining fee of $\$20$ plus $\$15$ per month, $m$. A rival gym charges no joining fee but $\$25$ per month. Write down an inequality in $m$ that must be satisfied for the first gym to be cheaper overall, and solve it.

   [3 marks]

   (b)

   The student can afford at most $\$140$ in total at the first gym. Write down an inequality for this constraint and solve it to find the maximum number of complete months, $m$, the student can afford.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A carpenter's weekly production of $x$ tables and $y$ chairs must satisfy $x \geq 0$, $y \geq 0$ and $x + 2y \leq 20$, based on available wood.

   (a)

   A region satisfying three inequalities represents feasible combinations of $x$ tables and $y$ chairs a carpenter can produce in a week: $x \geq 0$, $y \geq 0$, and $x + 2y \leq 20$. State, with a reason, whether the point $(4, 8)$ satisfies all three inequalities.

   [4 marks]

   (b)

   Whether the line $x + 2y = 20$ is drawn solid or broken depends on the type of inequality. Explain which type is used here and why, and state the type of line for $x+2y \leq 20$.

   [4 marks]

   (c)

   The carpenter earns a profit of $\$30$ per table and $\$20$ per chair. Given the constraint $x + 2y \leq 20$ (with $x, y \geq 0$ and both integers), find the combination of whole numbers of tables and chairs within this constraint that maximises profit, and state the maximum profit.

   [5 marks]

   Total for question 4: 13 marks

End of questions