Given:
The system of inequalities:
[tex]y\leq \dfrac{1}{2}x+2[/tex]
[tex]y<-2x-3[/tex]
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).
[tex]-2\leq \dfrac{1}{2}(-3)+2[/tex]
[tex]-2\leq -1.5+2[/tex]
[tex]-2\leq 0.5[/tex]
This statement is true.
[tex]-2<-2(-3)-3[/tex]
[tex]-2<6-3[/tex]
[tex]-2<3[/tex]
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).
[tex]2<-2(3)-3[/tex]
[tex]2<-6-3[/tex]
[tex]2<-9[/tex]
This statement is false because [tex]2>-9[/tex].
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.
