To determine if the point [tex]\((3, -2)\)[/tex] is in the solution set of the given system of linear inequalities, we need to check if it satisfies both inequalities:
1. [tex]\(y < -3\)[/tex]
2. [tex]\(y \leq \frac{2}{3}x - 4\)[/tex]
Let's evaluate the point [tex]\((3, -2)\)[/tex] for each inequality step by step.
### 1. Checking the inequality [tex]\(y < -3\)[/tex]:
Substitute the coordinates of the point [tex]\((3, -2)\)[/tex] into the inequality [tex]\(y < -3\)[/tex]:
- The [tex]\(y\)[/tex]-coordinate of the point is [tex]\(-2\)[/tex].
We need to check if:
[tex]\[
-2 < -3
\][/tex]
This statement is false since [tex]\(-2\)[/tex] is not less than [tex]\(-3\)[/tex].
### 2. Checking the inequality [tex]\(y \leq \frac{2}{3}x - 4\)[/tex]:
Substitute the coordinates of the point [tex]\((3, -2)\)[/tex] into the inequality [tex]\(y \leq \frac{2}{3}x - 4\)[/tex]:
- The [tex]\(x\)[/tex]-coordinate of the point is [tex]\(3\)[/tex], and the [tex]\(y\)[/tex]-coordinate is [tex]\(-2\)[/tex].
First, calculate the right-hand side of the inequality:
[tex]\[
\frac{2}{3} \cdot 3 - 4
\][/tex]
Perform the multiplication:
[tex]\[
2 - 4 = -2
\][/tex]
So, the inequality becomes:
[tex]\[
-2 \leq -2
\][/tex]
This statement is true since [tex]\(-2\)[/tex] is equal to [tex]\(-2\)[/tex], and thus it satisfies the inequality [tex]\(y \leq -2\)[/tex].
### Conclusion:
The point [tex]\((3, -2)\)[/tex] satisfies the second inequality [tex]\(y \leq \frac{2}{3}x - 4\)[/tex] but does not satisfy the first inequality [tex]\(y < -3\)[/tex]. Therefore, the point [tex]\((3, -2)\)[/tex] is in the solution set of the inequality:
[tex]\[
y \leq \frac{2}{3} x - 4
\][/tex]
but not in the solution set of the inequality:
[tex]\[
y < -3
\][/tex]
Hence, the point [tex]\((3, -2)\)[/tex] is in the solution set of:
[tex]\[
y \leq \frac{2}{3} x - 4
\][/tex]
