Question

Which system of inequalities does the graph represent:

2x + 7 is greater than or equal to 3 and y - x is greater than or equal to 2

2x +y is less than or equal to 3 and Y - X is greater than or equal to 2

2x + y is less than or equal to 3 and y - x is less than or equal to 2

2x + y is greater than or equal to 3 and y - x is less than or equal to 2

2x + y is greater than 3 + y - x is less than or equal to 2

Answer 1

Option 2x +y is less than or equal to 3 and Y - X is greater than or equal to 2

Explanation:

Locate on the graph a point that belongs to the region indicated, for example, the point (-3, 1)

Now we substitute this point in the inequalities and we test if it is satisfied. Let's try the second option.

We have:

`2x + y \leq 3\\\\ y - x \geq 2`

So:

`2 (-3) + (1) \leq 3\\-6 +1 \leq 3\\-5 \leq 3\\\\\ (1) - (-3) \geq 2\\(1) + 3 \geq 2\\4 \geq 2`

Both inequalities are satisfied. Therefore the correct option is option 2.

`2x + y \leq 3` and
`y - x \geq 2`

You can verify that by substituting this or some other point belonging to the region in the inequalities of the other options, these are not satisfied.

Answer 2

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

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