Question

Question was provided via image upload.

Explanation:

Step1: Analyze the first inequality \( y < -\frac{1}{2}x + 5 \)

The boundary line is \( y = -\frac{1}{2}x + 5 \), which has a slope of \( -\frac{1}{2} \) and a y - intercept of 5. Since the inequality is \( y < \), we shade below this line, and the line should be dashed (because it's a strict inequality, \( y
eq-\frac{1}{2}x + 5 \)).

Step2: Analyze the second inequality \( y\geq-\frac{1}{2}x - 1 \)

The boundary line is \( y = -\frac{1}{2}x - 1 \), which has a slope of \( -\frac{1}{2} \) and a y - intercept of - 1. Since the inequality is \( y\geq \), we shade above this line, and the line should be solid (because it's a non - strict inequality, \( y = -\frac{1}{2}x - 1 \) is included in the solution set).

Step3: Analyze the relationship between the two lines

The two lines \( y = -\frac{1}{2}x+5 \) and \( y = -\frac{1}{2}x - 1 \) are parallel (since they have the same slope, \( m = -\frac{1}{2} \)) and the region shaded below \( y = -\frac{1}{2}x + 5 \) and the region shaded above \( y=-\frac{1}{2}x - 1 \) will overlap (because the line \( y = -\frac{1}{2}x+5 \) is above the line \( y=-\frac{1}{2}x - 1 \) for all x - values).

Now let's analyze each option:

• Option A: The graph does not seem to represent the correct shading for the two inequalities.
• Option B: The graph does not represent the correct shading for the two parallel lines with the correct regions.
• Option C: The two parallel lines with the region between them (the overlapping region of the two shaded areas) is the solution set. The line \( y = -\frac{1}{2}x+5 \) is dashed (for \( y<-\frac{1}{2}x + 5 \)) and \( y=-\frac{1}{2}x - 1 \) is solid (for \( y\geq-\frac{1}{2}x - 1 \)), and the overlapping region is the set of points that satisfy both inequalities.
• Option D: Since the two lines are parallel and the shaded regions overlap, there are solutions.

Answer:

C (the option with the description "infinite number of solutions" and the corresponding graph)