Question

Use the image to answer the question.



Which point on the graph is a solution to the simultaneous inequalities:

y≤1/6x−6


y>2/3x−6


(1 point)
Responses

(3, 6)
(3, 6)

(0, -6)
(0, -6)

(2, −12)
(2, −12)

(−10, −10)
(−10, −10)

Answer 1

To determine which point on the graph is a solution to the simultaneous inequalities:

1. \( y \leq \frac{1}{6} x - 6 \)
2. \( y > \frac{2}{3} x - 6 \)

we need to check each of the given points:

1. **Point (3, 6)**:
- For the first inequality: \( 6 \leq \frac{1}{6}(3) - 6 \) → \( 6 \leq 0.5 - 6 \) → \( 6 \leq -5.5 \) (False)
- For the second inequality: \( 6 > \frac{2}{3}(3) - 6 \) → \( 6 > 2 - 6 \) → \( 6 > -4 \) (True)

Since (3, 6) does not satisfy the first inequality, it's **not a solution**.

2. **Point (0, -6)**:
- For the first inequality: \( -6 \leq \frac{1}{6}(0) - 6 \) → \( -6 \leq -6 \) (True)
- For the second inequality: \( -6 > \frac{2}{3}(0) - 6 \) → \( -6 > -6 \) (False)

Since (0, -6) does not satisfy the second inequality, it's **not a solution**.

3. **Point (2, −12)**:
- For the first inequality: \( -12 \leq \frac{1}{6}(2) - 6 \) → \( -12 \leq \frac{1}{3} - 6 \) → \( -12 \leq -5.67 \) (False)
- For the second inequality: \( -12 > \frac{2}{3}(2) - 6 \) → \( -12 > \frac{4}{3} - 6 \) → \( -12 > -4.33 \) (False)

Since (2, -12) does not satisfy the first inequality, it's **not a solution**.

4. **Point (−10, −10)**:
- For the first inequality: \( -10 \leq \frac{1}{6}(-10) - 6 \) → \( -10 \leq -\frac{10}{6} - 6 \) → \( -10 \leq -1.67 - 6 \) → \( -10 \leq -7.67 \) (True)
- For the second inequality: \( -10 > \frac{2}{3}(-10) - 6 \) → \( -10 > -\frac{20}{3} - 6 \) → \( -10 > -6.67 - 6 \) → \( -10 > -12.67 \) (True)

Since (−10, −10) satisfies both inequalities, it **is a solution** to the simultaneous inequalities.

Therefore, the correct answer is:

**(−10, −10)**.