Answer:Then the system of inequalities that represents the graph is second option:
2x +y is less than or equal to 3, and y-x is greater than or equal to 2
Step-by-step explanation: The area shaded is below the blue line and above the red line.
1.) The blue line goes through the points:
P1=(0,3)=(x1,y1)→x1=0, y1=3
P2=(1,1)=(x2,y2)→x2=1, y2=1
Slope: m=(y2-y1)/(x2-x1)
m=(1-3)/(1-0)→m=(-2)/(1)→m=-2
Point-slope form:
y-y1=m(x-x1)
y-3=-2(x-0)→y-3=-2(x)→y-3=-2x→y-3+3=-2x+3→y=-2x+3
The area below the blue line is:
y<=-2x+3→y+2x<=-2x+3→2x+y<=3
2x +y is less than or equal to 3
2.) The red line goes through the points:
P1=(0,2)=(x1,y1)→x1=0, y1=2
P2=(-2,0)=(x2,y2)→x2=-2, y2=0
Slope: m=(y2-y1)/(x2-x1)
m=(0-2)/(-2-0)→m=(-2)/(-2)→m=1
Point-slope form:
y-y1=m(x-x1)
y-2=1(x-0)→y-2=1(x)→y-2=x→y-2+2=x+2→y=x+2
The area above the red line is:
y>=x+2→y-x>=x+2-x→y-x>=2
y-x is greater than or equal to 2
Then the system of inequalities that represents the graph is:
2x +y is less than or equal to 3, and y-x is greater than or equal to 2
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