Question

Determine whether the points (–3,–2) and (3,2) are in the solution set of the system of inequalities below. y ≤ ½x + 2 y < –2x – 3

Answer 1

Given:

The system of inequalities:

`y\leq (1)/(2)x+2`

`y<-2x-3`

To find:

Whether the points (–3,–2) and (3,2) are in the solution set of the given system of inequalities.

Solution:

A point is in the solution set of the given system of inequalities if it satisfies both inequalities.

Check for the point (-3,-2).

`-2\leq (1)/(2)(-3)+2`

`-2\leq -1.5+2`

`-2\leq 0.5`

This statement is true.

`-2<-2(-3)-3`

`-2<6-3`

`-2<3`

This statement is also true.

Since the point (-3,-2) satisfies both inequalities, therefore (-3,-2) is in the solution set of the given system of inequalities.

Now, check for the point (3,2).

`2<-2(3)-3`

`2<-6-3`

`2<-9`

This statement is false because
`2>-9`.

Since the point (3,2) does not satisfy the second inequality, therefore (3,2) is not in the solution set of the given system of inequalities.