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:

$y\leq \dfrac{1}{2}x+2$

$y$

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).

$-2\leq \dfrac{1}{2}(-3)+2$

$-2\leq -1.5+2$

$-2\leq 0.5$

This statement is true.

$-2$

$-2$

$-2$

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).

$2$

$2$

$2$

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.