Given the inequality $3(n-6)<2(n+12)$, determine which integer makes the inequality false.
The given inequality is \(3(n - 6) < 2(n + 12)\). We want to determine which of the provided integers \(n\) makes this inequality false. The original inequality is true if the left-hand side is strictly less than the right-hand side. It is false if the left-hand side is greater than or equal to the right-hand side. We will simplify the expression first to find the range of values for which the inequality holds.
We expand the expression on both sides of the inequality:
Left-hand side (LHS): \(3(n - 6) = 3n - 18\)
Right-hand side (RHS): \(2(n + 12) = 2n + 24\)
The inequality becomes:
\(3n - 18 < 2n + 24\)
To solve for \(n\), we subtract \(2n\) from both sides:
\(n - 18 < 24\)
Then, we add \(18\) to both sides:
\(n < 42\)
This means the inequality holds true for all values of \(n\) strictly less than \(42\). Conversely, the inequality is false if \(n \geq 42\).
We evaluate the provided options: \(\{-5\}, \{3\}, \{12\}, \{42\}\).
Therefore, the integer that makes the inequality false is \(42\).
In real-world scenarios, inequalities help define boundaries, such as in production planning where a factory must ensure their production rate, say \(n\), does not exceed a certain capacity. For example, if you are planning to produce items at a rate \(n\) per hour, and your equipment efficiency follows the rule \(3(n - 6) < 2(n + 12)\), you need to find the limit of your production rate before the system becomes inefficient or false. Solving for \(n < 42\) tells you that the system works efficiently for any rate below 42 items per hour, and you'll know exactly at what point you need to stop or recalibrate.