Verified Answer :

  • Identify the given point \((x, y) = (1, 5)\) and the inequality \(y < 4x + 1\).
  • Substitute \(x = 1\) and \(y = 5\) into the inequality: \(5 < 4(1) + 1\).
  • Evaluate the expression: \(5 < 5\).
  • Conclude that the statement is false, so \((1, 5)\) is not a solution: \(\boxed{\text{no}}\).

Explanation

  1. Understanding the Problem and Data Analysis

We are examining the inequality \(y < 4x + 1\) to determine if the point \((1, 5)\) satisfies it. A point \((x, y)\) is a solution to an inequality if, upon substituting its coordinates into the inequality, the resulting mathematical statement is true. In this case, we have \(x = 1\) and \(y = 5\).

  1. Substitution and Evaluation

Substitute the values \(x = 1\) and \(y = 5\) into the inequality \(y < 4x + 1\):
\begin{align}
5 &< 4(1) + 1 \)
5 &< 4 + 1 \)
5 &< 5
\end{align}

  1. Conclusion

We compare the left-hand side and the right-hand side of the inequality. The result of our calculation is \(5 < 5\). Since 5 is equal to 5, the statement "5 is strictly less than 5" is false. Therefore, the point \((1, 5)\) does not satisfy the given inequality.

Examples

Inequalities are used to define regions in a plane, often representing constraints in real-world scenarios. For example, if a machine can produce a maximum of \(4x + 1\) items where \(x\) is the number of hours worked, and you want to know if producing 5 items in 1 hour is possible, you check if \(5 < 4(1) + 1\). If the inequality is false, it means you have reached or exceeded the capacity, and the production target isn't strictly within the limit. Understanding these boundaries helps in logistical planning and optimization.

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