Question

which graph shows the solution to the system of linear inequalities?
x + 5y ≥ 5
y ≤ 2x + 4

Explanation:

Step1: Analyze \( x + 5y \geq 5 \)

Rewrite it as \( y \geq -\frac{1}{5}x + 1 \). The boundary line \( y = -\frac{1}{5}x + 1 \) has a slope of \( -\frac{1}{5} \) and y - intercept at \( (0,1) \). Since the inequality is \( \geq \), the line is solid and we shade above it.

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

The boundary line \( y = 2x + 4 \) has a slope of \( 2 \) and y - intercept at \( (0,4) \). Since the inequality is \( \leq \), the line is solid and we shade below it.

Step3: Match with graphs

• For the line \( y = -\frac{1}{5}x + 1 \) (red line), it should be solid (all options seem to have solid red lines here). The slope is negative, so it goes down from left to right.
• For the line \( y = 2x + 4 \) (blue line), it has a positive slope (goes up from left to right) and y - intercept at \( (0,4) \). We need to shade above the red line and below the blue line.

Looking at the options, the fourth graph (the one with the blue line going up with positive slope and the red line with negative slope, and the shaded region that is above the red line and below the blue line) matches. Wait, no, let's re - check. Wait, the second inequality is \( y\leq2x + 4 \), so the blue line should have a positive slope. Let's check the slopes again. The line \( y = 2x+4 \) has a positive slope (rise over run is positive). The first three graphs have blue lines with negative slopes (except the fourth). Wait, no, in the fourth graph, the blue line has a positive slope (going up from left to right), the red line has a negative slope (going down from left to right). The shaded region should be above the red line (\( y\geq-\frac{1}{5}x + 1 \)) and below the blue line (\( y\leq2x + 4 \)). Let's check the y - intercepts. The red line has y - intercept at \( (0,1) \), the blue line in the fourth graph has y - intercept at \( (0,2) \)? Wait, no, \( y = 2x + 4 \) has y - intercept at \( (0,4) \). Wait, maybe I made a mistake in the graph labels. Wait, looking at the options again, the fourth graph (the one with the blue line going up, starting from below the red line's y - intercept? No, maybe I messed up. Wait, let's re - express the inequalities.

Wait, let's take a test point for \( x + 5y\geq5 \), say \( (0,0) \): \( 0+0 = 0\geq5 \) is false, so we shade away from \( (0,0) \), i.e., above the line \( x + 5y = 5 \). For \( y\leq2x + 4 \), test point \( (0,0) \): \( 0\leq0 + 4 \) is true, so we shade towards \( (0,0) \), i.e., below the line \( y = 2x+4 \).

Now, looking at the graphs:

• The blue line with positive slope (since \( y = 2x + 4 \) has positive slope) should be the one with positive slope. Among the four graphs, the fourth graph has a blue line with positive slope (going up from left to right), the red line with negative slope (going down from left to right). The shaded region in the fourth graph is above the red line (since when we test \( (0,2) \) for the red line: \( 0+5\times2=10\geq5 \), which is true, so \( (0,2) \) is in the region for \( x + 5y\geq5 \)) and below the blue line (test \( (0,2) \) in \( y\leq2x + 4 \): \( 2\leq0 + 4 \), which is true).

Answer:

The fourth graph (the one with the blue line having a positive slope and the red line having a negative slope, with the shaded region that is above the red line and below the blue line)