Determine which integer will make the inequality $12>2 x+4$ true.
We are tasked with solving the inequality \(12 > 2x + 4\) for \(x\) and determining which of the provided sets contains the integer that satisfies this condition. The set options are \(S = \{8\}\), \(S = \{4\}\), \(S = \{12\}\), and \(S = \{-2\}\). To solve the inequality, we need to isolate the variable \(x\).
To isolate \(x\), we first subtract 4 from both sides of the inequality:
\[12 - 4 > 2x\]
\[8 > 2x\]
Next, divide both sides by 2:
\[\frac{8}{2} > \frac{2x}{2}\]
\[4 > x\]
This means the inequality \(12 > 2x + 4\) is equivalent to \(x < 4\). Any integer \(x\) that is strictly less than 4 will satisfy the original inequality.
Now, we test each candidate set to see which one satisfies \(x < 4\):
Thus, the only set containing an integer that makes the inequality true is \(S = \{-2\}\).
Understanding inequalities is fundamental in budgeting. For example, if you have a budget of \(\$12\) and you need to pay a fixed fee of \(\$4\), the remaining amount you can spend is represented by the inequality \(12 > 2x + 4\), where \(2x\) is the cost of items you want to buy. By solving \(x < 4\), you quickly determine that you can afford at most 3 items, as buying 4 items would cost exactly \(\$12\), reaching your limit. This application helps in making informed financial decisions daily.