To determine if each ordered pair is a solution to the system of linear inequalities given by:
- \( y < x - 5 \)
- \( y \leq -2x + 1 \)
we will test each ordered pair \((x, y)\) to see if it satisfies both inequalities.
Testing Each Ordered Pair
-
For (0, -10):
- First inequality: \( -10 < 0 - 5 \) → \( -10 < -5 \) (True)
- Second inequality: \( -10 \leq -2(0) + 1 \) → \( -10 \leq 1 \) (True)
- Result: (0, -10) is a solution.
-
For (8, 1):
- First inequality: \( 1 < 8 - 5 \) → \( 1 < 3 \) (True)
- Second inequality: \( 1 \leq -2(8) + 1 \) → \( 1 \leq -16 + 1 \) → \( 1 \leq -15 \) (False)
- Result: (8, 1) is NOT a solution.
-
For (5, 0):
- First inequality: \( 0 < 5 - 5 \) → \( 0 < 0 \) (False)
- Second inequality (not needed since first is false, but let's check): \( 0 \leq -2(5) + 1 \) → \( 0 \leq -10 + 1 \) → \( 0 \leq -9 \) (False)
- Result: (5, 0) is NOT a solution.
-
For (2, 5):
- First inequality: \( 5 < 2 - 5 \) → \( 5 < -3 \) (False)
- Second inequality (not needed since first is false, but let's check): \( 5 \leq -2(2) + 1 \) → \( 5 \leq -4 + 1 \) → \( 5 \leq -3 \) (False)
- Result: (2, 5) is NOT a solution.
-
For (1, -6):
- First inequality: \( -6 < 1 - 5 \) → \( -6 < -4 \) (True)
- Second inequality: \( -6 \leq -2(1) + 1 \) → \( -6 \leq -2 + 1 \) → \( -6 \leq -1 \) (False)
- Result: (1, -6) is NOT a solution.
Summary of Results
- (0, -10) is a solution.
- (8, 1) is NOT a solution.
- (5, 0) is NOT a solution.
- (2, 5) is NOT a solution.
- (1, -6) is NOT a solution.
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