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Determine which integer will make the inequality $3(n+1)>5(n-3)$ true.


A$S: \{9\}$
B$S: \{0\}$
C$S: \{15\}$
D$S: \{18\}$

Verified Answer :

  • Simplify the given inequality \(3(n+1) > 5(n-3)\) to \(3n + 3 > 5n - 15\).
  • Isolate \(n\) to find the range \(n < 9\).
  • Evaluate each provided set to identify which contains an integer satisfying the condition.
  • Conclude that only the set \(\{0\}\) contains a valid integer, as \(\boxed{\{0\}}\).

Explanation

  1. Understanding the Problem and Data Analysis

We are given the linear inequality \(3(n+1) > 5(n-3)\) and need to determine which of the sets {9}, {0}, {15}, and {18} contain an integer \(n\) that makes the inequality true. The inequality involves basic algebraic operations such as expansion, subtraction, and comparison of values. Our approach is to first isolate \(n\) to find its range and then test the given values.

  1. Solving the Inequality for n

To solve for \(n\), we first expand both sides of the inequality:
\[3(n+1) > 5(n-3)\]
\[3n + 3 > 5n - 15\]
Next, we rearrange the terms to isolate \(n\) on one side:
\[3 + 15 > 5n - 3n\]
\[18 > 2n\]
Dividing both sides by 2, we obtain:
\[9 > n\]
This means the inequality \(3(n+1) > 5(n-3)\) is satisfied for all values of \(n\) such that \(n < 9\).

  1. Evaluating the Provided Options

Now we check each candidate set provided:

  • For the set {9}: \(n = 9\). Substituting into the inequality \(9 < 9\) gives a false statement, so 9 does not satisfy the inequality.
  • For the set {0}: \(n = 0\). Substituting into the inequality \(0 < 9\) gives a true statement, so 0 satisfies the inequality.
  • For the set {15}: \(n = 15\). Substituting into the inequality \(15 < 9\) gives a false statement, so 15 does not satisfy the inequality.
  • For the set {18}: \(n = 18\). Substituting into the inequality \(18 < 9\) gives a false statement, so 18 does not satisfy the inequality.

Thus, the only set that contains a value of \(n\) satisfying the inequality is {0}.

Examples

In real-life scenarios, such as project management or logistics, we often deal with constraints where resources must remain below a certain limit to maintain efficiency. For example, if a machine can handle a workload defined by a specific mathematical inequality, finding the range of acceptable inputs allows you to optimize operations without exceeding capacity. Here, solving for \(n < 9\) is analogous to ensuring that a production batch size \(n\) stays within a safe operational threshold.

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