liner inequalities

First one...here's one way to do this....pick a point in the shaded area...I'll choose (1,2)  = (x , y)

Note that this satisfies   y  ≥   1      and    y  - x  >  0

So...the second answer is correct for the first one

Similarly...for the second one...let's pick  ( 4 , 0)  = (x , y)

Look  at the last answer..... plug in  4 for x   and 0  for y  and we have

0 > -2(4) - 2

0 > -8 - 2

0 >  -10     which is true

Also

0  ≤ -(1/2)(4) + 6

0 ≤ -2 + 6

0 ≤ 4     which is also true

Lastly

0 ≤ 4     which is also true

Since  all  three inequalities are true, this must be the correct answer

Check for yourself that at least one equation in each of  the other three answers gives an incorrect inequality

For example in the first two answers

y ≤ -2x + 6     will not be true for  (x, y)  =  (4,0)

And in the third answer

y ≥ x  will not be true  for  (x, y)  = ( 4,0)

Hope that helps.....!!!!