Systems of Inequations — practice questions

Draw the region satisfying $3x + 4y < 24$ with $x \geq 0$ and $y \geq 0$. State the intercepts, the line style, and how you decided which side to shade.

A shaded region on a graph has a solid boundary line passing through $(0, 5)$ and $(10, 0)$, and the shaded area lies above the line. Write the inequation, and explain how each feature of the graph determined a feature of your answer.

Explain why the inequality sign reverses when both sides are multiplied by a negative number, and use your reasoning to explain why testing a point is more reliable than rearranging an inequation into the form $y \leq \dots$ before graphing.

Write the system of inequations for this situation: a delivery van carries boxes weighing 20 kg and crates weighing 35 kg. The van can carry at most 700 kg and at most 25 items in total. At least 5 crates must be carried.

A region is defined by $x + y \leq 10$, $2x + y \leq 16$, $x \geq 0$, $y \geq 2$. Find all corner points of the feasible region, verifying each, and describe what each corner represents if $x$ and $y$ are hours spent on two tasks.

A caterer must prepare at least 60 servings using two dishes. Dish A serves 4 people and takes 20 minutes; dish B serves 6 people and takes 45 minutes. Only 6 hours of preparation time is available, and at most 12 dishes can fit in the kitchen. Determine whether the caterer's requirements can all be met, and analyse how the answer changes if the serving requirement rises.