- Identify that the inequality \(\frac{5}{6}m - 4 < \frac{1}{3}m + 5\) is false when \(\frac{5}{6}m - 4 \geq \frac{1}{3}m + 5\).
- Simplify the negated inequality to find that it is true for \(m \geq 18\).
- Evaluate each element in the set \(\{-18, 2, 15, 20\}\) to see which satisfies \(m \geq 18\).
- Conclude that only \(20\) satisfies the condition, thus the final answer is \(\boxed{\{20\}}\).
Explanation
- Problem Analysis
We are given the set \(S = \{-18, 2, 15, 20\}\) and the inequality \(\frac{5}{6}m - 4 < \frac{1}{3}m + 5\). The goal is to determine which elements of the set \(S\) make this inequality false. An inequality is false when the condition is not met, which corresponds to the negation of the original statement. Thus, we are looking for values of \(m\) that satisfy \(\frac{5}{6}m - 4 \geq \frac{1}{3}m + 5\).
- Solving the Inequality
To find the range of \(m\) that makes the original inequality false, we solve the negated inequality \(\frac{5}{6}m - 4 \geq \frac{1}{3}m + 5\).
First, subtract \(\frac{1}{3}m\) from both sides:
\[\frac{5}{6}m - \frac{1}{3}m - 4 \geq 5\]
Convert \(\frac{1}{3}\) to \(\frac{2}{6}\) to combine the terms:
\[\left(\frac{5}{6} - \frac{2}{6}\right)m - 4 \geq 5\]
\[\frac{3}{6}m - 4 \geq 5\]
\[\frac{1}{2}m - 4 \geq 5\]
Next, add 4 to both sides:
\[\frac{1}{2}m \geq 9\]
Multiply both sides by 2:
\[m \geq 18\]
Thus, the inequality \(\frac{5}{6}m - 4 < \frac{1}{3}m + 5\) is false if and only if \(m \geq 18\).
- Testing Elements and Conclusion
Now, we test each element in the set \(S = \{-18, 2, 15, 20\}\) against the condition \(m \geq 18\):
- For \(m = -18\): \(-18 \geq 18\) is false.
- For \(m = 2\): \(2 \geq 18\) is false.
- For \(m = 15\): \(15 \geq 18\) is false.
- For \(m = 20\): \(20 \geq 18\) is true.
Therefore, only \(m = 20\) makes the original inequality false. This corresponds to the option \(S = \{20\}\).
Examples
In resource management, an inequality like this can represent a threshold. Suppose \(\frac{5}{6}m - 4\) represents the output of a production line and \(\frac{1}{3}m + 5\) represents the operating cost. If you are checking if your production is always less than the cost, finding the values of \(m\) for which the inequality is false helps identify when the production might become too costly (i.e., when cost is no longer greater than production). Understanding these thresholds is essential for optimizing efficiency and planning budgets in various industrial or personal finance scenarios.