What system of inequalities is represented by the graph?

Option C is correct.

Step-by-step explanation:

We are given with a graph with shaded region.

from the graph a line passes through point ( -2 , 0 ) and another line passes through point ( 0 , -2 ). using these point first we find equation of lines.

Also, from graph region shade by both lines is in opposite direction of point origin that is ( 0 , 0 ) which means for coordinates of origin does not satisfy the inequalities.

We need to find inequalities.

Option A).

y ≥ x - 2 (1)

y ≥ -2x - 6 (2)

put x = -2 & y = 0 in any inequality.

in (1),

LHS = y = 0

RHS = x - 2 = -2 - 2 = -4

LHS ≠ RHS

now, in (2)

LHS = y = 0

RHS = -3x - 6 = -3(-2) - 6 = 6 - 6 = 0

LHS = RHS

put x = 0 & y = -2 in (1)

LHS = y = -2

RHS = x - 2 = 0 - 2 = -2

LHS = RHS

⇒ y = x - 2 &  y = -3x - 6 are our required equation of line.

Now we check are these our required inequalities.

By putting x = 0 and y = 0

in (1) we get,

0 ≥ 0 -2  ⇒ 0 ≥ -2

in (2) we get,

0 ≥ -3(0) - 6 ⇒ 0 ≥ -6

In both inequality origin satisfies them.

Thus this pair of inequality is not our answer.

Since Points from graph satisfy these y = x - 2 & y = -3x - 6

So, Option B and D is reject.

Now Option C).

y ≤ x - 2 (3)

y ≤ -2x - 6 (4)

These set of equations of lines are our required ones.

So, We only check if origin satisfies these inequality or not.

By putting x = 0 and y = 0

in (3) we get,

0 ≥ 0 -2  ⇒ 0 ≤ -2

in (4) we get,

0 ≥ -3(0) - 6 ⇒ 0 ≤ -6

In both inequality origin does not satisfies them.

Thus this pair of inequality is our answer.

Therefore, Option C is correct.