Question

Please help!

Which of the following points lie in the solution set to the following system of inequalities?y < −3x + 3y < x + 2
(1, −5)
(1, 5)
(5, 1)
(−1, 5)

Answer 1

Plug the values and check if they return a true statement or not:

Plugging
`(x,y) = (1,-5)` we have

`-5 < -3 -15 < 1 + 2 \iff -5 < -18 < 3` which is false

Plugging
`(x,y) = (1,5)` we have

`5 < -3 +15 < 1 + 2 \iff 5 < 12 < 3` which is false

Plugging
`(x,y) = (5, 1)` we have

`1 < -15 + 3 < 5 + 2 \iff 1 < -12 < 7` which is false

Plugging
`(x,y) = (-1, 5)` we have

`5 < 3 +15 < -1 + 2 \iff 5 < 18 < 1` which is false

So, all of the options seem wrong.

Answer 2

(1,-5)

Explanation:

The given system of inequalities are;

`y\:<\:-3x+3`

and

`y\:<\:x+2`

The point that is a solution will satisfy the two inequalities.

Checking for (1,-5)

`-5\:<\:-3(1)+3` and
`-5\:<\:1+2`

`-5\:<\:0`: True
`-5\:<\:3` : True

This point lies in the solution region.

Checking for (1,5)

`5\:<\:-3(1)+3` and
`5\:<\:1+2`

`5\:<\:0`: False
`5\:<\:3` : False

This point does not lie in the solution region.

Checking for (5,1)

`1\:<\:-3(5)+3` and
`1\:<\:5+2`

`1\:<\:-12`: False
`1\:<\:7` : True

This point does not lie in the solution region.

Checking for (-1,5)

`5\:<\:-3(-1)+3` and
`5\:<\:-1+2`

`5\:<\:6`: True
`5\:<\:1` : False

This point does not lie in the solution region.