Verified Answer :

  • Substitute \(x = 4\) and \(y = 8\) into the expression \(8x - 4y\).
  • Calculate the result: \(8(4) - 4(8) = 32 - 32 = 0\).
  • Compare the result to \(-16\): \(0 > -16\) is a true statement.
  • Conclude that \((4, 8)\) is a solution: \(\boxed{\text{yes}}\).

Explanation

  1. Analyzing the Problem

To determine if the ordered pair \((4, 8)\) is a solution to the inequality \(8x - 4y > -16\), we need to substitute the given values \(x = 4\) and \(y = 8\) into the left-hand side of the inequality. A pair \((x, y)\) is a solution if the resulting statement is true.

  1. Substitution and Calculation

Substitute \(x = 4\) and \(y = 8\) into the expression \(8x - 4y\):
\[8(4) - 4(8) = 32 - 32 = 0.\]

  1. Conclusion

Now compare the result to the right-hand side of the inequality, which is \(-16\):
\[0 > -16.\]
Since \(0\) is greater than \(-16\), the inequality holds true. Therefore, the ordered pair \((4, 8)\) is indeed a solution to the inequality.

Examples

In real-life scenarios, inequalities are used to define boundaries for acceptable or optimal conditions. For instance, if you have a budget of \(\$16\) and you are managing costs where \(x\) and \(y\) represent different types of expenses, you might use an inequality to ensure you stay within limits. In this specific case, testing if a certain set of values satisfies a condition helps verify if your budget or production plan is feasible, just as we verified if the point \((4, 8)\) satisfies the given boundary constraint.

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