Answer :
To determine which system of linear inequalities includes the point \((3, -2)\) in its solution set, let's evaluate whether the point satisfies each given inequality step-by-step.
First, consider the inequality:
[tex]\[ y < -3 \][/tex]
Plugging in the coordinates of the point \((3, -2)\):
[tex]\[ -2 < -3 \][/tex]
This statement is false because \(-2\) is not less than \(-3\).
Next, consider the inequality:
[tex]\[ y \leq \frac{2}{3} x - 4 \][/tex]
Plugging in the coordinates of the point \((3, -2)\):
[tex]\[ -2 \leq \frac{2}{3} \cdot 3 - 4 \][/tex]
Simplify the right-hand side:
[tex]\[ -2 \leq 2 - 4 \][/tex]
[tex]\[ -2 \leq -2 \][/tex]
This statement is true because \(-2\) is indeed less than or equal to \(-2\).
To conclude, the point \((3, -2)\) satisfies the second inequality \( y \leq \frac{2}{3}x - 4 \) but does not satisfy the first inequality \( y < -3 \).
So, the point \((3, -2)\) is in the solution set of the system represented by the second inequality:
[tex]\[ y \leq \frac{2}{3} x - 4 \][/tex]
First, consider the inequality:
[tex]\[ y < -3 \][/tex]
Plugging in the coordinates of the point \((3, -2)\):
[tex]\[ -2 < -3 \][/tex]
This statement is false because \(-2\) is not less than \(-3\).
Next, consider the inequality:
[tex]\[ y \leq \frac{2}{3} x - 4 \][/tex]
Plugging in the coordinates of the point \((3, -2)\):
[tex]\[ -2 \leq \frac{2}{3} \cdot 3 - 4 \][/tex]
Simplify the right-hand side:
[tex]\[ -2 \leq 2 - 4 \][/tex]
[tex]\[ -2 \leq -2 \][/tex]
This statement is true because \(-2\) is indeed less than or equal to \(-2\).
To conclude, the point \((3, -2)\) satisfies the second inequality \( y \leq \frac{2}{3}x - 4 \) but does not satisfy the first inequality \( y < -3 \).
So, the point \((3, -2)\) is in the solution set of the system represented by the second inequality:
[tex]\[ y \leq \frac{2}{3} x - 4 \][/tex]
