- The goal is to identify which linear inequality holds true for four given points: \((-4, -1)\), \((-2, 4)\), \((3, -3)\), and \((3, -4)\).
- Testing each point against the candidate inequalities reveals that the first, third, and fourth options fail for at least one point.
- Specifically, the point \((3, -3)\) satisfies \(y \le -2x + 3\) because \(-3 = -3\), while it fails the strict inequality \(y < -2x + 3\).
- All points satisfy the inequality \(\boxed{y \le -2x + 3}\).
Explanation
- Problem Analysis and Strategy
To determine which inequality represents the given table of values, we must test each provided inequality with every coordinate pair \((x, y)\) in the table. The points are: \((-4, -1)\), \((-2, 4)\), \((3, -3)\), and \((3, -4)\). An inequality is valid only if it holds true for every single point.
- Testing Each Inequality Option
We evaluate each option one by one:
Option 1: \(y < -2x + 3\)
- For \((-4, -1)\): \(-1 < -2(-4) + 3 = 8 + 3 = 11\). (True)
- For \((-2, 4)\): \(4 < -2(-2) + 3 = 4 + 3 = 7\). (True)
- For \((3, -3)\): \(-3 < -2(3) + 3 = -6 + 3 = -3\). This statement \(-3 < -3\) is false.
Option 2: \(y \le -2x + 3\)
- For \((-4, -1)\): \(-1 \le -2(-4) + 3 = 11\). (True)
- For \((-2, 4)\): \(4 \le -2(-2) + 3 = 7\). (True)
- For \((3, -3)\): \(-3 \le -2(3) + 3 = -3\). (True, since \(-3 = -3\))
- For \((3, -4)\): \(-4 \le -2(3) + 3 = -3\). (True, since \(-4 < -3\))
Since all points satisfy this inequality, this is a strong candidate.
Option 3: \(y > -\frac{1}{2}x - 3\)
- For \((-4, -1)\): \(-1 > -\frac{1}{2}(-4) - 3 = 2 - 3 = -1\). This statement \(-1 > -1\) is false.
Option 4: \(y \le -\frac{1}{2}x - 3\)
- For \((-2, 4)\): \(4 \le -\frac{1}{2}(-2) - 3 = 1 - 3 = -2\). This statement \(4 \le -2\) is false.
- Conclusion
Based on the testing above, the only inequality that is satisfied by all four given points (\((-4, -1)\), \((-2, 4)\), \((3, -3)\), and \((3, -4)\)) is \(y \le -2x + 3\). Thus, this is the correct choice.
Examples
Linear inequalities are used in real-world scenarios like supply chain management to determine if a production plan fits within resource constraints. For example, if a factory has a limited budget and can produce two types of items, \(x\) and \(y\), at specific costs, the inequality \(2x + y \le 3\) might represent the constraint that production must remain within a set capacity. By checking if various production combinations (the points) satisfy this boundary, planners can ensure their operations stay within safe and profitable limits.