Question

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Explanation:

Formulate the inequality

Let \(x\) represent the number of bags of topsoil and \(y\) represent the number of bags of mulch.

$$ 40x + 20y \le 480 $$

Find the boundary intercepts

Find the intercepts of the boundary line \(40x + 20y = 480\):

• For \(x = 0\):

$$ 20y = 480 \implies y = 24 $$

• For \(y = 0\):

$$ 40x = 480 \implies x = 12 $$

Identify the correct graph representation

The correct graph must show:

• A solid boundary line passing through the \(y\)-intercept \((0, 24)\) and the \(x\)-intercept \((12, 0)\).
• Shading below and to the left of the boundary line in the first quadrant, since \(x \ge 0\), \(y \ge 0\), and the test point \((0,0)\) satisfies \(40(0) + 20(0) = 0 \le 480\).
• The visible graph option shows a boundary line with an \(x\)-intercept at \(12\) and a \(y\)-intercept at \(24\), with the shaded region representing the feasible combinations.

Answer:

The correct graph is the one representing the inequality \(40x + 20y \le 480\), where \(x\) is the number of bags of topsoil and \(y\) is the number of bags of mulch.

This graph is characterized by:

• A solid boundary line with a \(y\)-intercept at \((0, 24)\) and an \(x\)-intercept at \((12, 0)\).
• Shading in the first quadrant below and to the left of this boundary line.

• (A) Solid line with intercepts at (12, 0) and (0, 24), shaded region below the line in the first quadrant (Correct answer)
• (B) Solid line with intercepts at (24, 0) and (0, 12), shaded region below the line in the first quadrant