Question

which graph shown the solution to this system of inequalities?
y < \\(\frac{1}{2}x - 2\\)
y ≤ -2x + 4

a. graph a
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b. graph b
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c. graph c
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d. graph d
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Explanation:

Step1: Analyze \( y < \frac{1}{2}x - 2 \)

The inequality \( y < \frac{1}{2}x - 2 \) has a dashed line (since it's \( < \), not \( \leq \)) with slope \( \frac{1}{2} \) and y - intercept \( - 2 \). The region below this line is shaded.

Step2: Analyze \( y \leq - 2x + 4 \)

The inequality \( y \leq - 2x + 4 \) has a solid line (since it's \( \leq \)) with slope \( - 2 \) and y - intercept \( 4 \). The region below or on this line is shaded.

Step3: Find the intersection region

We need to find the region that is below the dashed line \( y=\frac{1}{2}x - 2 \) and below or on the solid line \( y = - 2x+4 \).

Let's check the lines:

• For \( y=\frac{1}{2}x - 2 \), when \( x = 0 \), \( y=-2 \); when \( y = 0 \), \( x = 4 \).
• For \( y=-2x + 4 \), when \( x = 0 \), \( y = 4 \); when \( y=0 \), \( x = 2 \).

Now, let's analyze the options:

• Option A: The dashed line is \( y=\frac{1}{2}x - 2 \) (correct slope and intercept, dashed) and the solid line is \( y=-2x + 4 \) (correct slope and intercept, solid). The shaded region is below both lines, which matches the solution of the system.
• Option B: The shaded region is above the dashed line \( y=\frac{1}{2}x - 2 \), which does not satisfy \( y < \frac{1}{2}x - 2 \).
• Option C: The dashed line does not seem to have the correct slope or intercept for \( y=\frac{1}{2}x - 2 \) (the line in C for \( y=\frac{1}{2}x - 2 \) - like line does not pass through (4,0) or (0, - 2) correctly).
• Option D: The shaded region is above the dashed line \( y=\frac{1}{2}x - 2 \), which does not satisfy \( y < \frac{1}{2}x - 2 \).

Answer:

A