Determine which integer will make the inequality $3(n+1)>5(n-3)$ true.
The objective is to identify which integer from the given options \(S = \{9\}, S = \{0\}, S = \{15\}, S = \{18\}\) satisfies the linear inequality \(3(n+1) > 5(n-3)\). We will proceed by simplifying the inequality step-by-step to isolate \(n\).
We begin by distributing the values inside the parentheses:
\[3 \times n + 3 \times 1 > 5 \times n - 5 \times 3\]
\[3n + 3 > 5n - 15\]
Next, we rearrange the inequality to collect the terms involving \(n\) on one side and the constant values on the other. Subtracting \(3n\) from both sides gives:
\[3 > 2n - 15\]
Adding 15 to both sides yields:
\[18 > 2n\]
Finally, dividing both sides by 2 results in:
\[9 > n\]
Thus, any integer \(n\) that is less than 9 satisfies the original inequality.
Now we evaluate the given options against the condition \(n < 9\):
Since 0 is the only integer in our set that is less than 9, it is the only option that satisfies the inequality.
This type of inequality is useful in resource management, such as determining the number of items or hours of work you can afford. For example, if you have a budget of \$18 for a service that costs \$2 per unit (plus some overhead), you can calculate the maximum units \(n\) you can purchase using an inequality like \(2n < 18\). Solving such inequalities helps in planning and making cost-effective decisions in everyday situations.