Question

y ≥ -\frac{1}{4}x + 3\
x > -2\
\
a) image of a graph\
b) image of a graph\
c) image of a graph\
d) image of a graph

Explanation:

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

The line \( y = -\frac{1}{2}x + 2 \) has a slope of \( -\frac{1}{2} \) and y - intercept 2. The inequality \( y \geq -\frac{1}{2}x + 2 \) means we shade above the line (since \( y \) is greater than or equal to the line's value), and the line should be solid (because of the "≥" sign).

Step2: Analyze \( x > - 2 \)

The inequality \( x > - 2 \) means we shade to the right of the vertical dashed line \( x=-2 \) (dashed because \( x > - 2 \) is a strict inequality, not including \( x = - 2 \)).

Step3: Match with graphs

• For the line \( y=-\frac{1}{2}x + 2 \), we look for a solid line with negative slope.
• For \( x > - 2 \), we look for shading to the right of \( x=-2 \).

Now let's check the options:

• Option a: The line \( y = -\frac{1}{2}x+2 \) (solid, negative slope) and shading to the right of \( x = - 2 \) (vertical dashed line at \( x=-2 \)) and above the line. This matches our requirements.
• Option b: The lines have positive slopes, which do not match \( y = -\frac{1}{2}x + 2 \)'s negative slope. So eliminate.
• Option c: The vertical line and the line's slope and shading do not match the required conditions.
• Option d: The shading and line characteristics do not match.

Answer:

A. (The graph labeled 'a')