Question

45 which graph shows the solution to the inequality $y \leq \frac{1}{2}x + 2$? a graph a b graph b answer choices c and d are on the following page.

Explanation:

Step1: Analyze the inequality

The inequality is \( y \leq \frac{1}{2}x + 2 \). First, identify the boundary line: \( y=\frac{1}{2}x + 2 \). The slope \( m=\frac{1}{2} \) and y - intercept \( b = 2 \). Since the inequality is \( \leq \), the line should be solid (not dashed) and we shade below the line.

Step2: Check the y - intercept

For the line \( y=\frac{1}{2}x + 2 \), when \( x = 0 \), \( y=2 \). So the line should cross the y - axis at \( (0,2) \).

Step3: Analyze Graph A and Graph B

• Graph A: Check the y - intercept. It seems to cross the y - axis at a point other than \( (0,2) \). Also, the slope and shading direction need to be checked.
• Graph B: The line crosses the y - axis at \( (0,2) \) (matches the y - intercept of \( y=\frac{1}{2}x + 2 \)). The slope of the line in Graph B: Let's take two points. From the graph, when \( x=- 4 \), \( y = 0 \) (since \( 0=\frac{1}{2}(-4)+2=-2 + 2=0 \)) and when \( x = 0 \), \( y = 2 \). The slope between \( (-4,0) \) and \( (0,2) \) is \( \frac{2 - 0}{0-(-4)}=\frac{2}{4}=\frac{1}{2} \), which matches the slope of the line \( y=\frac{1}{2}x + 2 \). Also, since the inequality is \( y\leq\frac{1}{2}x + 2 \), we shade below the line. Graph B shows shading below the line (the lower part relative to the line \( y=\frac{1}{2}x + 2 \)).

Answer:

B