Answered

Determine which integer will make the inequality $12 > 2x + 4$ true.


A$S: \{8\}$
B$S: \{4\}$
C$S: \{12\}$
D$S: \{-2\}$

Verified Answer :

  • Simplify the given inequality \(12 > 2x + 4\) by subtracting 4 and dividing by 2 to obtain \(x < 4\).
  • Test each candidate from the provided sets:
  • For \(x=8\): \(8 < 4\) is False.
  • For \(x=4\): \(4 < 4\) is False.
  • For \(x=12\): \(12 < 4\) is False.
  • For \(x=-2\): \(-2 < 4\) is True.
  • The set containing the correct integer is \(\{-2\}\), which satisfies the inequality.
  • The final answer is \(\boxed{\{-2\}}\).

Explanation

  1. Problem Analysis

We are given the inequality \(12 > 2x + 4\) and a collection of sets containing potential integer candidates for \(x\). Our goal is to identify which set contains an integer that makes this inequality true. The sets provided are \(S = \{8\}\), \(S = \{4\}\), \(S = \{12\}\), and \(S = \{-2\}\).

  1. Simplifying the Inequality

To simplify the inequality \(12 > 2x + 4\), we subtract 4 from both sides:
\[12 - 4 > 2x\]
\[8 > 2x\]
Next, divide both sides by 2:
\[\frac{8}{2} > x\]
\[4 > x\]
Thus, any integer \(x\) that satisfies the original inequality must be strictly less than 4.

  1. Evaluating Candidates

Now, we evaluate each candidate set to see if its element satisfies the inequality \(x < 4\):

  • For \(S = \{8\}\): Is \(8 < 4\)? This is false.
  • For \(S = \{4\}\): Is \(4 < 4\)? This is false, as 4 is not strictly less than 4.
  • For \(S = \{12\}\): Is \(12 < 4\)? This is false.
  • For \(S = \{-2\}\): Is \(-2 < 4\)? This is true, as -2 is indeed less than 4.

Therefore, the set that contains an integer satisfying the inequality is \(S = \{-2\}\).

  1. Conclusion

Based on our analysis, only the integer \(-2\) satisfies the condition \(x < 4\), which corresponds to the original inequality \(12 > 2x + 4\). The correct set is \(\{-2\}\). \(\boxed{\{-2\}}\)

Examples

Inequalities are fundamental in real-life resource management and budgeting. For instance, if you have a budget of \$12 for a party and you know that your fixed costs are \$4 and each guest costs you \$2, you can model your spending as \(4 + 2x < 12\), where \(x\) is the number of guests. By solving this inequality, you can determine the maximum number of guests you can invite to stay within your budget. Understanding how to solve these linear inequalities ensures you can make smart financial decisions in everyday situations.

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