Question

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Explanation:

Formulate the system of inequalities

Let \(x\) represent the number of bags of chips (horizontal axis) and \(y\) represent the number of candy bars (vertical axis).
Based on the problem description:

• Stephen wants to buy at least 8 items:

$$x + y \ge 8$$

• Stephen has at most \$12 in his pocket, where chips cost \$2.25 each and candy bars cost \$1.50 each:

$$2.25x + 1.50y \le 12$$

Find the boundary line intercepts

For the first boundary line \(x + y = 8\):

• \(x\)-intercept: \((8, 0)\)
• \(y\)-intercept: \((0, 8)\)
• Shading is above the line (\(\ge\)).

For the second boundary line \(2.25x + 1.50y = 12\):

• \(x\)-intercept: \(\frac{12}{2.25} = 5.33 \implies (5.33, 0)\)
• \(y\)-intercept: \(\frac{12}{1.50} = 8 \implies (0, 8)\)
• Shading is below the line (\(\le\)).

Identify the correct graph

• The boundary lines intersect on the \(y\)-axis at \((0, 8)\).
• The region must be above \(x + y = 8\) and below \(2.25x + 1.50y = 12\).
• This forms a narrow triangular region bounded between the two lines in the first quadrant, starting from \((0, 8)\) and extending down to the \(x\)-axis between \(x = 5.33\) and \(x = 8\).
• Looking at the options, the first graph shows this exact shaded region between the two boundary lines.

Answer:

• (A) The first graph, showing the shaded region bounded between the two lines with a shared y-intercept at (0, 8) (Correct answer)
• (B) The second graph, showing an unshaded gap between the two boundary lines
• (C) The third graph, showing the shaded region extending infinitely to the upper right
• (D) The fourth graph, showing a different shaded region