Question

Solve the system of inequalities.
3x − 2y >10

−6x + 4y ≤16

(1 point)
Responses

(-4,-5)
(-4,-5)

(3,-10)
(3,-10)

(0,-5)

Answer 1

To solve the system of inequalities, we need to analyze each inequality separately and then find the solutions that satisfy both.

1. **First inequality: \( 3x - 2y > 10 \)**

Rearranging the inequality gives:
\[
-2y > -3x + 10
\]
Dividing through by -2 (and reversing the inequality):
\[
y < \frac{3}{2}x - 5
\]

2. **Second inequality: \( -6x + 4y \leq 16 \)**

Rearranging this inequality yields:
\[
4y \leq 6x + 16
\]
Dividing through by 4 (and maintaining the inequality):
\[
y \leq \frac{3}{2}x + 4
\]

Now we have the two inequalities:
1. \( y < \frac{3}{2}x - 5 \)
2. \( y \leq \frac{3}{2}x + 4 \)

Next, we can test the points given in the responses to see which one satisfies both inequalities.

- **Testing point (-4, -5)**:
1. \( y < \frac{3}{2}(-4) - 5 \) → \( -5 < -6 - 5 \) → \( -5 < -11 \) (False)
2. \( y \leq \frac{3}{2}(-4) + 4 \) → \( -5 \leq -6 + 4 \) → \( -5 \leq -2 \) (False)

- **Testing point (3, -10)**:
1. \( y < \frac{3}{2}(3) - 5 \) → \( -10 < 4.5 - 5 \) → \( -10 < -0.5 \) (True)
2. \( y \leq \frac{3}{2}(3) + 4 \) → \( -10 \leq 4.5 + 4 \) → \( -10 \leq 8.5 \) (True)

- **Testing point (0, -5)**:
1. \( y < \frac{3}{2}(0) - 5 \) → \( -5 < -5 \) (False)
2. \( y \leq \frac{3}{2}(0) + 4 \) → \( -5 \leq 4 \) (True)

From the testing, only the point (3, -10) satisfies both inequalities.

Hence, the solution to the system of inequalities is:
\[
(3, -10)
\]