To determine which integer [tex]\( n \)[/tex] makes the inequality [tex]\( 6(n-5) < 3(n+4) \)[/tex] true, let's solve the inequality step by step:
1. Start with the given inequality:
[tex]\[
6(n-5) < 3(n+4)
\][/tex]
2. Distribute the constants inside the parentheses:
[tex]\[
6n - 30 < 3n + 12
\][/tex]
3. To isolate [tex]\( n \)[/tex], we need to get all the [tex]\( n \)[/tex]-terms on one side and the constant terms on the other. Subtract [tex]\( 3n \)[/tex] from both sides:
[tex]\[
6n - 3n - 30 < 12
\][/tex]
Simplify:
[tex]\[
3n - 30 < 12
\][/tex]
4. Next, add 30 to both sides to isolate the [tex]\( n \)[/tex]-term:
[tex]\[
3n - 30 + 30 < 12 + 30
\][/tex]
Simplify:
[tex]\[
3n < 42
\][/tex]
5. Finally, divide both sides by 3 to solve for [tex]\( n \)[/tex]:
[tex]\[
\frac{3n}{3} < \frac{42}{3}
\][/tex]
Simplify:
[tex]\[
n < 14
\][/tex]
Therefore, the inequality [tex]\( 6(n-5) < 3(n+4) \)[/tex] is true for values of [tex]\( n \)[/tex] less than 14.
Now, we check the given integers:
- [tex]\( 11 \)[/tex]: [tex]\( 11 < 14 \)[/tex] (True)
- [tex]\( 14 \)[/tex]: [tex]\( 14 < 14 \)[/tex] (False)
- [tex]\( 30 \)[/tex]: [tex]\( 30 < 14 \)[/tex] (False)
- [tex]\( 42 \)[/tex]: [tex]\( 42 < 14 \)[/tex] (False)
Thus, the integer from the given set that satisfies the inequality is [tex]\( 11 \)[/tex].
