Verified Answer :

  • Substitute the point \((x, y) = (2, 5)\) into the expression \(3x + 2y\).
  • Calculate the result: \(3(2) + 2(5) = 16\).
  • Compare the result with the inequality condition: \(16 \geq 16\).
  • Conclude that the point satisfies the inequality, so the answer is \(\boxed{\text{a}}\).

Explanation

  1. Problem Analysis

We are given a point in the Cartesian plane with coordinates \((x, y) = (2, 5)\) and an inequality \(3x + 2y \geq 16\). To determine if the point satisfies the inequality, we need to substitute the values of \(x\) and \(y\) from the point into the expression on the left-hand side of the inequality.

  1. Evaluation of the Expression

Substitute \(x = 2\) and \(y = 5\) into the linear expression \(3x + 2y\):
\[3(2) + 2(5)\]
Perform the multiplications:
\[3 \times 2 = 6\]
\[2 \times 5 = 10\]
Add the products together:
\[6 + 10 = 16\]

  1. Comparison and Conclusion

Now, compare the calculated result with the value specified in the inequality:
The calculated value is \(16\), and the inequality requires the result to be greater than or equal to \(16\).
Since \(16 \geq 16\) is true, the point \((2, 5)\) satisfies the given inequality. Therefore, the answer is 'yes'.

Examples

In geography, inequalities are often used to define regions on a map. For example, if a company wants to find all locations within a certain distance of a supply center to optimize delivery routes, they might use inequalities to represent boundary lines on a coordinate grid. By checking if specific delivery coordinates (like (2, 5)) fall within the region defined by an inequality (like \(3x + 2y \geq 16\)), businesses can efficiently determine if a location is serviced or requires a different route.

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