1. 1
   A theme park sets ride restrictions using inequalities. One ride requires a rider's age, $x$ years, to satisfy $3x-2<10$. A second ride requires the double inequality $1<x\leq5$ for a different scaled quantity $x$. A third ride uses a quadratic condition $x^2\leq25$ for a safety measurement $x$.

   (a)

   Solve the inequality $3x-2<10$.

   [1 mark]

   • A$x<3$
   • B$x<4$
   • C$x>4$
   • D$x<12$

   (b)

   For the second ride, which integer values satisfy $1<x\leq5$?

   [1 mark]

   • A$2,3,4$
   • B$1,2,3,4,5$
   • C$2,3,4,5$
   • D$1,2,3,4$

   (c)

   Solve $x^2\leq25$ for the third ride's safety condition.

   [1 mark]

   • A$-5<x<5$
   • B$x\leq5$
   • C$x\leq-5$ or $x\geq5$
   • D$-5\leq x\leq5$

   Total for question 1: 3 marks

2. 2
   A parking garage offers a reduced rate for customers whose parking time $x$ hours satisfies $2x+5\leq19$. A second sign restricts vehicle height, $y$ metres, using $3(y-1)>2y+4$.

   (a)

   A parking garage charges according to the number of hours parked, $x$. A sign states that customers pay a reduced rate only if $2x+5\leq19$. Solve this inequality to find the range of values of $x$ for the reduced rate.

   [2 marks]

   (b)

   A second sign restricts vehicle height, $y$ metres, using $3(y-1)>2y+4$. Solve this inequality to find the range of values of $y$ not permitted to enter.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A delivery company accepts packages with weight $w$ kilograms satisfying $2w-3\geq5$ and $w<10$. The company also tracks a batch of oversized packages whose dimensions satisfy the quadratic inequality $w^2-w-12>0$.

   (a)

   A delivery company requires that a package's weight, $w$ kilograms, satisfies both $2w-3\geq5$ and $w<10$. Solve each inequality separately, then write the combined solution as a single inequality.

   [3 marks]

   (b)

   For oversized packages, the dimension $w$ must satisfy $w^2-w-12>0$. Solve this quadratic inequality, showing your critical values and stating the full solution set.

   [3 marks]

   Total for question 3: 6 marks

4. 4
   A factory's quality control department sets two independent inequality conditions on component length $x$ cm: $4x-7\leq2x+9$ and $\frac{x}{3}+2>5$. The factory also monitors a batch where the length must satisfy the quadratic inequality $x^2+3x+2>0$, and needs to describe the region on a coordinate grid satisfying $y\leq2x+1$ for a related design constraint.

   (a)

   A factory produces components with length $x$ cm. Quality control requires $4x-7\leq2x+9$ and separately $\frac{x}{3}+2>5$. Solve both inequalities and represent the combined solution set on a number line, describing clearly which endpoints are open or closed.

   [4 marks]

   (b)

   For a batch monitored separately, the length $x$ must satisfy $x^2+3x+2>0$. Solve this quadratic inequality fully, stating your critical values clearly.

   [4 marks]

   (c)

   For a related design constraint on a coordinate grid, describe fully (without drawing) the region defined by $y\leq2x+1$, stating which side of the line $y=2x+1$ is included and whether the boundary line itself is part of the region.

   [5 marks]

   Total for question 4: 13 marks

End of questions