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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 \(3(n + 1) > 5(n - 3)\) by expanding both sides to get \(3n + 3 > 5n - 15\).
  • Isolate the variable \(n\) to find that the inequality is equivalent to \(n < 9\).
  • Test each provided set to identify which one contains an integer satisfying the condition \(n < 9\).
  • Conclude that the set \(S = \{0\}\) is the correct choice because \(0 < 9\). The final answer is \(\boxed{S = \{0\}}\).

Explanation

  1. Problem Analysis

We are given the inequality \(3(n + 1) > 5(n - 3)\) and asked to determine which of the sets \(S = \{9\}, S = \{0\}, S = \{15\}, S = \{18\}\) contains an integer \(n\) that satisfies this inequality. The process involves simplifying the given expression to isolate the variable \(n\) and then checking which option matches the resulting condition.

  1. Solving the Inequality

Starting with the inequality \(3(n + 1) > 5(n - 3)\), we expand the terms on both sides:
\[3n + 3 > 5n - 15\]
Next, we rearrange the inequality to isolate \(n\). We subtract \(3n\) from both sides:
\[3 > 2n - 15\]
Then, we add 15 to both sides:
\[18 > 2n\]
Finally, dividing both sides by 2, we obtain:
\[9 > n\]
This means the inequality holds true for any integer \(n\) that is strictly less than 9.

  1. Checking the Sets

Now we evaluate each of the provided sets to see which contains an integer \(n\) such that \(n < 9\):

  • For \(S = \{9\}\): \(9 < 9\) is false.
  • For \(S = \{0\}\): \(0 < 9\) is true.
  • For \(S = \{15\}\): \(15 < 9\) is false.
  • For \(S = \{18\}\): \(18 < 9\) is false.

The only set that satisfies the condition is \(S = \{0\}\). Thus, the correct choice is the set containing the integer 0.

Examples

Consider a scenario where you are setting a limit for a computer program's memory usage. If a variable \(n\) represents the number of tasks currently running and the system constraint is described by \(3(n+1) > 5(n-3)\), solving this inequality helps you determine the safe limit for \(n\). If the current state of your system is \(n=0\) tasks, the inequality holds, indicating that your configuration is currently within the operational parameters. Understanding these inequalities helps in managing resources efficiently and ensuring system stability.

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