- Evaluate each given coordinate pair against both inequalities in the system.
- Test the point \((-6, 4)\): \(4 < -\frac{3}{2}(-6) - 2\) simplifies to \(4 < 7\) (True), and \(-6 + 3(4) \ge 6\) simplifies to \(6 \ge 6\) (True).
- Since both conditions hold, the point \((-6, 4)\) satisfies the system of inequalities.
- Final Answer: \(\boxed{(-6, 4)}\).
Explanation
- Problem Analysis
To determine which of the provided points \((-2, 1)\), \((0, 2)\), \((-6, 4)\), or \((-4, 2)\) satisfies the system of inequalities:
\[ \begin{cases} y < -\frac{3}{2}x - 2 \ x + 3y \geq 6 \end{cases} \]
we will test each point by substituting its coordinates \((x, y)\) into both inequalities and checking if they hold true. A point is a solution only if it satisfies both inequalities simultaneously.
- Testing the Points
Let's test each point one by one:
1. Point \((-2, 1)\):
- Inequality 1: \(y < -\frac{3}{2}x - 2 \implies 1 < -\frac{3}{2}(-2) - 2 \implies 1 < 3 - 2 \implies 1 < 1\). This is false.
2. Point \((0, 2)\):
- Inequality 1: \(y < -\frac{3}{2}x - 2 \implies 2 < -\frac{3}{2}(0) - 2 \implies 2 < -2\). This is false.
3. Point \((-6, 4)\):
- Inequality 1: \(y < -\frac{3}{2}x - 2 \implies 4 < -\frac{3}{2}(-6) - 2 \implies 4 < 9 - 2 \implies 4 < 7\). This is true.
- Inequality 2: \(x + 3y \geq 6 \implies -6 + 3(4) \geq 6 \implies -6 + 12 \geq 6 \implies 6 \geq 6\). This is true.
Since both inequalities are satisfied, \((-6, 4)\) is a solution.
4. Point \((-4, 2)\):
- Inequality 1: \(y < -\frac{3}{2}x - 2 \implies 2 < -\frac{3}{2}(-4) - 2 \implies 2 < 6 - 2 \implies 2 < 4\). This is true.
- Inequality 2: \(x + 3y \geq 6 \implies -4 + 3(2) \geq 6 \implies -4 + 6 \geq 6 \implies 2 \geq 6\). This is false.
- Conclusion
Based on our testing, only the point \((-6, 4)\) satisfies both inequalities in the system. Therefore, \((-6, 4)\) is the correct answer.
Examples
This method of testing points is essential in linear programming, a field used to optimize outcomes in business and logistics. For instance, if a company wants to minimize production costs (represented by one inequality) while meeting minimum output requirements (represented by another), they can visualize or test specific production strategies (points) to see if they fall within the 'feasible region' that satisfies all constraints simultaneously. Mastering this technique allows for efficient resource management and better decision-making in real-world scenarios.