Answer :
To determine whether the point [tex]\((3, -2)\)[/tex] lies within the solution set of the given system of linear inequalities:
1. First inequality: [tex]\( y < -3 \)[/tex]
- We substitute the point [tex]\((3, -2)\)[/tex] into the inequality:
[tex]\[ y = -2 \\ -2 < -3 \][/tex]
- This statement is false because [tex]\(-2\)[/tex] is not less than [tex]\(-3\)[/tex].
2. Second inequality: [tex]\( y \leq \frac{2}{3} x - 4 \)[/tex]
- We substitute the point [tex]\((3, -2)\)[/tex] into the inequality:
[tex]\[ y = -2 \quad \text{and} \quad x = 3 \\ -2 \leq \frac{2}{3} \cdot 3 - 4 \][/tex]
- Calculating the right-hand side:
[tex]\[ \frac{2}{3} \cdot 3 = 2 \\ 2 - 4 = -2 \][/tex]
- Therefore, the inequality becomes:
[tex]\[ -2 \leq -2 \][/tex]
- This statement is true because [tex]\(-2\)[/tex] is equal to [tex]\(-2\)[/tex].
Combining the results, the point [tex]\((3, -2)\)[/tex] does not satisfy the first inequality [tex]\( y < -3 \)[/tex] but does satisfy the second inequality [tex]\( y \leq \frac{2}{3} x - 4 \)[/tex].
Therefore, the point [tex]\((3, -2)\)[/tex] lies 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].
1. First inequality: [tex]\( y < -3 \)[/tex]
- We substitute the point [tex]\((3, -2)\)[/tex] into the inequality:
[tex]\[ y = -2 \\ -2 < -3 \][/tex]
- This statement is false because [tex]\(-2\)[/tex] is not less than [tex]\(-3\)[/tex].
2. Second inequality: [tex]\( y \leq \frac{2}{3} x - 4 \)[/tex]
- We substitute the point [tex]\((3, -2)\)[/tex] into the inequality:
[tex]\[ y = -2 \quad \text{and} \quad x = 3 \\ -2 \leq \frac{2}{3} \cdot 3 - 4 \][/tex]
- Calculating the right-hand side:
[tex]\[ \frac{2}{3} \cdot 3 = 2 \\ 2 - 4 = -2 \][/tex]
- Therefore, the inequality becomes:
[tex]\[ -2 \leq -2 \][/tex]
- This statement is true because [tex]\(-2\)[/tex] is equal to [tex]\(-2\)[/tex].
Combining the results, the point [tex]\((3, -2)\)[/tex] does not satisfy the first inequality [tex]\( y < -3 \)[/tex] but does satisfy the second inequality [tex]\( y \leq \frac{2}{3} x - 4 \)[/tex].
Therefore, the point [tex]\((3, -2)\)[/tex] lies 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].
