# Which system of inequalities is represented by the graph?

A. y<=x-5 and y>=-3x+2
B. y<=x-5 and y>=-3x-2
C. y<=-x+5 and y>=-3x-2
D. y<=x+5 and y>=-3x-2

## No graph can be shown here by students.

All choices consist of intersections of regions, so
let's look at one of the choices:

B. y<=x-5 and y>=-3x-2

I expect to see the region below y = x-5, with the line shown as a solid line
intersected with the region above y = -3x - 2, with the line shown as a solid
line.

## Well, I like to think of it as a game of "find the right answer." And you know what they say, finding the right answer is like finding a needle in a haystack, except the haystack is on fire, and the needle is made of glitter. So let me sprinkle some humor into your question.

Now, looking at the graph, it seems like we're dealing with a love triangle. One of these inequalities represents the person who wants to be closer to the point (5,2), while the other person wants to move away from it. So let's see which option fits this drama.

When we examine the possible answers, it becomes clear that the only system of inequalities that fits the bill is option C, y ≤ -x + 5 and y ≥ -3x - 2. So, grab some popcorn and get ready for some mathematical drama!

## To determine which system of inequalities is represented by the graph, we need to analyze the slopes and y-intercepts of the lines shown.

Looking at the graph, we can see that the lines have a negative slope and pass through the point (0, 5). Let's compare these characteristics to the options provided.

A. The first inequality, y <= x-5, represents a line with a positive slope and a y-intercept of -5. This does not match the graph.
B. The first inequality, y <= x-5, represents a line with a positive slope and a y-intercept of -5. This does not match the graph.
C. The first inequality, y <= -x+5, represents a line with a negative slope and a y-intercept of 5. This matches the graph. The second inequality, y >= -3x-2, represents a line with a negative slope and a y-intercept of -2. This also matches the graph. Therefore, this option represents the correct system of inequalities.
D. The second inequality, y >= -3x-2, represents a line with a negative slope and a y-intercept of -2. This matches the graph. However, the first inequality, y <= x+5, represents a line with a positive slope and a y-intercept of 5. This does not match the graph.

Therefore, the system of inequalities represented by the graph is option C: y <= -x+5 and y >= -3x-2.

## To determine which system of inequalities is represented by the graph, we can refer to the slope-intercept form of the inequalities (y = mx + b).

Let's analyze the given options using the given graph:

Option A: y <= x - 5 and y >= -3x + 2
To check if this option is correct, let's identify the slope and y-intercept of each equation:
- For y <= x - 5: slope = 1, y-intercept = -5
- For y >= -3x + 2: slope = -3, y-intercept = 2

Looking at the graph, the line with a slope of 1 and a y-intercept of -5 seems accurate. However, the line with a slope of -3 and a y-intercept of 2 does not match the graph. So option A is not correct.

Option B: y <= x - 5 and y >= -3x - 2
To assess this option, let's determine the slope and y-intercept:
- For y <= x - 5: slope = 1, y-intercept = -5
- For y >= -3x - 2: slope = -3, y-intercept = -2

Comparing the lines in option B to the graph, we can observe that the line matching the slope of 1 and a y-intercept of -5 aligns with the graph. Furthermore, the line with a slope of -3 and a y-intercept of -2 is also consistent with the graph. Hence, option B is correct.

Option C: y <= -x + 5 and y >= -3x - 2
Examining the slopes and y-intercepts:
- For y <= -x + 5: slope = -1, y-intercept = 5
- For y >= -3x - 2: slope = -3, y-intercept = -2

The lines represented by option C do not match the graph. Therefore, option C is incorrect.

Option D: y <= x + 5 and y >= -3x - 2
Analyzing the slopes and y-intercepts:
- For y <= x + 5: slope = 1, y-intercept = 5
- For y >= -3x - 2: slope = -3, y-intercept = -2

While the line with a slope of 1 and a y-intercept of 5 is consistent with the graph, the line with a slope of -3 and a y-intercept of -2 does not match the graph. Consequently, option D is not correct.

In conclusion, the graph represents the system of inequalities:
B. y <= x - 5 and y >= -3x - 2