Verified Answer :

  • Identify the coordinates \(x=3\) and \(y=-7\).
  • Substitute these values into the inequality \(y \geq x-10\), resulting in \(-7 \geq 3-10\).
  • Simplify the expression to get \(-7 \geq -7\), which is true.
  • Conclude that the point \((3,-7)\) satisfies the inequality: \(\boxed{\text{yes}}\).

Explanation

  1. Analyzing the Problem and Data

We are given an ordered pair \((x, y) = (3, -7)\) and a linear inequality \(y \geq x - 10\). Our objective is to determine whether the given point satisfies this inequality by substituting the coordinate values into the expression.

  1. Substituting and Evaluating the Inequality

First, we identify the values for our variables from the coordinate pair:
\(x = 3\)
\(y = -7\)
Next, we substitute these values into the given inequality \(y \geq x - 10\).
Substituting \(x = 3\) and \(y = -7\) gives us:
\(-7 \geq 3 - 10\)
Now, we simplify the right side of the inequality:
\(3 - 10 = -7\)
So, the inequality becomes:
\(-7 \geq -7\)

  1. Concluding the Verification

We compare the left and right sides of the inequality: \(-7 \geq -7\). Since \(-7\) is equal to \(-7\), the inequality holds true because the definition of \(\geq\) (greater than or equal to) includes equality. Therefore, the ordered pair \((3, -7)\) satisfies the given inequality.

Examples

This type of evaluation is fundamental in coordinate geometry and linear programming. For example, if you are determining whether a specific point (representing a resource allocation, like 3 hours of labor and -7 units of cost) falls within a feasible region defined by a constraint (like a production budget), testing the coordinate ensures that your plan complies with the set limits. Understanding if a point satisfies an inequality allows you to map out valid zones in real-world planning and optimization problems.

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