- Evaluate each provided ordered pair against both inequalities: \(y \geq -0.5x\) and \(y < 0.5x + 1\).
- Discard any set that contains a pair failing either condition.
- Identify that the set \((5, -2), (3, 1), (4, 2)\) satisfies both conditions for every pair.
- The correct option is the third one: \(\boxed{(5, -2), (3, 1), (4, 2)}\).
Explanation
- Problem Analysis
We are given a system of two linear inequalities:
1) \(y \geq -\frac{1}{2}x\)
2) \(y < \frac{1}{2}x + 1\)
We need to identify which of the provided sets of three ordered pairs \((x, y)\) satisfies both inequalities for all pairs in the set. An ordered pair \((x, y)\) is a solution if it results in true statements for both inequality expressions simultaneously.
- Evaluating Ordered Pairs
Let us test each candidate pair \((x, y)\) for the given system of inequalities:
1) \(y \geq -0.5x\)
2) \(y < 0.5x + 1\)
Testing pair \((5, -2)\):
1) \(-2 \geq -0.5(5) = -2.5\) (True)
2) \(-2 < 0.5(5) + 1 = 3.5\) (True)
This pair is a solution.
Testing pair \((3, 1)\):
1) \(1 \geq -0.5(3) = -1.5\) (True)
2) \(1 < 0.5(3) + 1 = 2.5\) (True)
This pair is a solution.
Testing pair \((-4, 2)\):
1) \(2 \geq -0.5(-4) = 2\) (True)
2) \(2 < 0.5(-4) + 1 = -1\) (False)
This pair is not a solution.
Testing pair \((3, -1)\):
1) \(-1 \geq -0.5(3) = -1.5\) (True)
2) \(-1 < 0.5(3) + 1 = 2.5\) (True)
This pair is a solution.
Testing pair \((4, -3)\):
1) \(-3 \geq -0.5(4) = -2\) (False)
This pair is not a solution.
Testing pair \((4, 2)\):
1) \(2 \geq -0.5(4) = -2\) (True)
2) \(2 < 0.5(4) + 1 = 3\) (True)
This pair is a solution.
Testing pair \((-3, 1)\):
1) \(1 \geq -0.5(-3) = 1.5\) (False)
This pair is not a solution.
- Conclusion
Now we check the given options:
1) \((5, -2), (3, 1), (-4, 2)\): \((-4, 2)\) failed.
2) \((5, -2), (3, -1), (4, -3)\): \((4, -3)\) failed.
3) \((5, -2), (3, 1), (4, 2)\): All three are solutions.
4) \((5, -2), (-3, 1), (4, 2)\): \((-3, 1)\) failed.
The correct set of ordered pairs is \((5, -2), (3, 1), (4, 2)\).
Examples
Systems of inequalities like this are used in logistics and resource allocation. For example, if a delivery drone must operate within a specific air corridor defined by elevation constraints relative to its horizontal position, these inequalities determine the safe flying zone. Ensuring all planned coordinates fall within the defined bounds prevents collisions and ensures regulatory compliance.