To determine which point on the graph is a solution to the simultaneous inequalities:
- \( y \leq \frac{1}{6} x - 6 \)
- \( y > \frac{2}{3} x - 6 \)
we need to check each of the given points:
- 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.
- 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.
- 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.
- 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).