Determine which integer will make the inequality $x-5>12$ true.
The given inequality is \(x - 5 > 12\), where \(x\) is an integer. We are provided with four potential sets containing values for \(x\): \(\{17\}\), \(\{18\}\), \(\{12\}\), and \(\{16\}\). Our goal is to determine which of these sets contains an integer that satisfies the inequality. An inequality describes a relationship where one value is strictly greater than or less than another. To solve this, we must isolate \(x\) to find the range of values that make the expression true.
To isolate \(x\), we apply the addition property of inequality. We add 5 to both sides of the original inequality \(x - 5 > 12\):
\[x - 5 + 5 > 12 + 5\]
\[x > 17\]
This tells us that for the inequality to be true, \(x\) must be any integer strictly greater than 17.
Now, we evaluate each of the provided sets to see which integer satisfies the condition \(x > 17\):
The only integer that satisfies the condition is 18.
Based on our analysis, the only value that makes the inequality \(x - 5 > 12\) true is \(x = 18\). Therefore, the correct set is \(\{18\}\). The final answer is \(\boxed{18}\).
Linear inequalities are fundamental in real-world resource allocation. For instance, if you have a budget of \$12 to spend on art supplies after an initial fixed cost of \$5, the total cost of your materials \(x\) must satisfy the condition \(x - 5 > 12\) if you want to exceed your planned savings. Solving this reveals the minimum amount you would need to spend to surpass that threshold, helping you make informed financial decisions.