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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 :

  • Expand the given inequality: \(3n + 3 > 5n - 15\).
  • Isolate the variable \(n\): \(18 > 2n\), which simplifies to \(n < 9\).
  • Test each set option: only \(0 < 9\) is true.
  • The set containing the correct integer is \(\boxed{S=\{0\}}\).

Explanation

  1. Problem Analysis

We are tasked with solving the linear inequality \(3(n+1) > 5(n-3)\) for an integer \(n\) and determining which of the given sets \(S=\{9\}, S=\{0\}, S=\{15\}, S=\{18\}\) contains a value of \(n\) that makes the inequality a true statement. A linear inequality is an expression that uses the inequality symbols \(<\), \(>\), \(\leq\), or \(\geq\) and has variables of degree 1. We will simplify the expression by expanding both sides, isolating the variable \(n\), and then testing the provided options.

  1. Solving the Inequality

First, expand both sides of the given inequality:
\[3(n+1) > 5(n-3)\]
Distributing the constants gives:
\[3n + 3 > 5n - 15\]
Now, to isolate the \(n\) terms, subtract \(3n\) from both sides:
\[3 > 5n - 3n - 15\]
\[3 > 2n - 15\]
Next, add 15 to both sides to isolate the constant term on the left:
\[3 + 15 > 2n\]
\[18 > 2n\]
Finally, divide both sides by 2 to solve for \(n\):
\[9 > n\]
This means the inequality \(3(n+1) > 5(n-3)\) is true for all integers \(n\) less than 9.

  1. Testing the Options and Conclusion

Now we evaluate the given sets to see which contains an integer that satisfies \(n < 9\):

  • For \(S=\{9\}\): The value is 9. Since \(9 < 9\) is false, this set is not a solution.
  • For \(S=\{0\}\): The value is 0. Since \(0 < 9\) is true, this set is a solution.
  • For \(S=\{15\}\): The value is 15. Since \(15 < 9\) is false, this set is not a solution.
  • For \(S=\{18\}\): The value is 18. Since \(18 < 9\) is false, this set is not a solution.

Therefore, the only set containing an integer that satisfies the original inequality is \(S=\{0\}\).

Examples

Imagine you are setting a budget for a school event. If the cost of the event depends on the number of attendees \(n\), and you need to ensure that the total cost does not exceed a certain threshold, you might set up an inequality to determine the maximum number of people you can invite. Solving such inequalities helps organizers make informed decisions, ensuring the event stays within budget while still accommodating the maximum number of guests possible. This is a practical way to manage resources and plan events efficiently.

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