- Simplify the inequality \(\frac{5}{6}m - 4 < \frac{1}{3}m + 5\) to isolate \(m\), yielding \(m < 18\).
- Recognize that the inequality is false when \(m \geq 18\).
- Evaluate the set \(S = \{-18, 2, 15, 20\}\) to see which values satisfy \(m \geq 18\).
- Conclude that only \(20\) makes the inequality false, so the final answer is \(\boxed{20}\).
Explanation
- Understanding the Problem
The inequality given is \(\frac{5}{6}m - 4 < \frac{1}{3}m + 5\). We want to find which integer(s) \(m\) from the set \(S = \{-18, 2, 15, 20\}\) make this inequality false. An inequality is false when the condition is violated, meaning the opposite relationship holds true. In this case, \(\frac{5}{6}m - 4 < \frac{1}{3}m + 5\) is false if and only if \(\frac{5}{6}m - 4 \geq \frac{1}{3}m + 5\).
- Simplifying the Inequality
To determine which values make the inequality false, we first simplify the inequality to find the range of \(m\) that satisfies it.
Starting with:
\[\frac{5}{6}m - 4 < \frac{1}{3}m + 5\]
Subtract \(\frac{1}{3}m\) from both sides:
\[\frac{5}{6}m - \frac{1}{3}m - 4 < 5\]
Since \(\frac{1}{3} = \frac{2}{6}\), this becomes:
\[\frac{5}{6}m - \frac{2}{6}m - 4 < 5\]
\[\frac{3}{6}m - 4 < 5\]
\[\frac{1}{2}m - 4 < 5\]
Now, add 4 to both sides:
\[\frac{1}{2}m < 9\]
Multiply by 2:
\[m < 18\]
So, the original inequality is true for all \(m < 18\). Consequently, the original inequality is false when \(m \geq 18\).
- Evaluating the Values in the Set
We need to check which values from the set \(S = \{-18, 2, 15, 20\}\) satisfy the condition for the inequality to be false, which is \(m \geq 18\).
- For \(m = -18\): \(-18 < 18\), so the inequality is true.
- For \(m = 2\): \(2 < 18\), so the inequality is true.
- For \(m = 15\): \(15 < 18\), so the inequality is true.
- For \(m = 20\): \(20 \geq 18\), so the inequality is false.
Thus, only the value \(m = 20\) makes the inequality false.
Examples
This math problem is similar to setting limits in real-world scenarios. For example, if a machine operates efficiently only when its temperature \(T\) is below a certain threshold (e.g., \(T < 18^\circ\text{C}\)), any value \(T \geq 18^\circ\text{C}\) would represent a state where the efficiency condition is not met, or "false". Similarly, by solving inequalities, we can determine the safe operational range for manufacturing equipment, budgeting for personal expenses, or even optimizing shipping loads.