question 20 of 25
which of the following systems of inequalities would produce the region
indicated on the graph below?
graph omitted
a. ( y leq -x + 4 ); ( y > x + 2 ); ( y geq 0 ); ( x geq 0 )
b. ( y leq -x + 4 ); ( y leq x + 2 ); ( y geq 0 ); ( x geq 0 )
c. ( y leq -x + 4 ); ( y geq x + 2 ); ( y geq 0 ); ( x geq 0 )
d. ( y leq -x + 4 ); ( y < x + 2 ); ( y geq 0 ); ( x geq 0 )
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# Explanation:
## Step1: Analyze the line $ y = -x + 4 $
The line $ y=-x + 4 $ has a y - intercept of 4 and an x - intercept of 4. The shaded region is below this line, so the inequality should be $ y\leq -x + 4 $ (since the line is solid, we use $ \leq $).

## Step2: Analyze the line $ y=x + 2 $
The line $ y = x+2 $ has a y - intercept of 2 and a slope of 1. The shaded region is below this line. Since the line is dashed (from the graph's appearance, as the shaded region is below and the boundary is not included in the shaded area in terms of the inequality for this line), the inequality should be $ y<x + 2 $ (if the line was solid, we would use $ \leq $, but here it's dashed, so we use $ < $).

## Step3: Analyze the inequalities $ y\geq0 $ and $ x\geq0 $
The shaded region is in the first quadrant (since $ x\geq0 $ and $ y\geq0 $), so these two inequalities are part of the system.

Now let's check the options:
- Option A: $ y>x + 2 $ is incorrect because the shaded region is below $ y=x + 2 $.
- Option B: $ y\leq x + 2 $ is incorrect because the line $ y=x + 2 $ is dashed (so we should use $ < $ instead of $ \leq $).
- Option C: $ y\geq x + 2 $ is incorrect as the shaded region is below $ y=x + 2 $.
- Option D: $ y\leq -x + 4 $, $ y<x + 2 $, $ y\geq0 $, $ x\geq0 $ matches our analysis.

# Answer: D. $ y\leq -x + 4; y<x + 2; y\geq0; x\geq0 $

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QUESTION IMAGE

Question

Explanation:

Step1: Analyze the line \( y = -x + 4 \)

Step2: Analyze the line \( y=x + 2 \)

Step3: Analyze the inequalities \( y\geq0 \) and \( x\geq0 \)

Answer: