Answer:
Option C is correct.
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.