To solve the problem of determining which integer makes the given inequality [tex]\( 4(n - 5) < 3(n + 11) \)[/tex] false, we need to follow a detailed step-by-step process to understand why a particular integer does not satisfy the inequality.
### Step-by-Step Solution:
1. Expand and Simplify the Inequality:
Start by expanding both sides of the inequality:
[tex]\[
4(n - 5) < 3(n + 11)
\][/tex]
[tex]\[
4n - 20 < 3n + 33
\][/tex]
2. Isolate the Variable [tex]\( n \)[/tex]:
To isolate [tex]\( n \)[/tex], subtract [tex]\( 3n \)[/tex] from both sides:
[tex]\[
4n - 3n - 20 < 33
\][/tex]
Simplifying this, we get:
[tex]\[
n - 20 < 33
\][/tex]
3. Solve for [tex]\( n \)[/tex]:
Add 20 to both sides to solve for [tex]\( n \)[/tex]:
[tex]\[
n < 53
\][/tex]
This tells us that the inequality [tex]\( 4(n - 5) < 3(n + 11) \)[/tex] holds true for any value of [tex]\( n \)[/tex] less than 53.
4. Determine Which Integer Makes the Inequality False:
We need to check the given integers: 4, 11, 53, and -8 to determine which one makes the inequality false.
- For [tex]\( n = 4 \)[/tex]:
[tex]\[
4(4 - 5) < 3(4 + 11)
\][/tex]
[tex]\[
4(-1) < 3(15)
\][/tex]
[tex]\[
-4 < 45 \quad (\text{true})
\][/tex]
- For [tex]\( n = 11 \)[/tex]:
[tex]\[
4(11 - 5) < 3(11 + 11)
\][/tex]
[tex]\[
4(6) < 3(22)
\][/tex]
[tex]\[
24 < 66 \quad (\text{true})
\][/tex]
- For [tex]\( n = 53 \)[/tex]:
[tex]\[
4(53 - 5) < 3(53 + 11)
\][/tex]
[tex]\[
4(48) < 3(64)
\][/tex]
[tex]\[
192 < 192 \quad (\text{false})
\][/tex]
This is false because 192 is not less than 192.
- For [tex]\( n = -8 \)[/tex]:
[tex]\[
4(-8 - 5) < 3(-8 + 11)
\][/tex]
[tex]\[
4(-13) < 3(3)
\][/tex]
[tex]\[
-52 < 9 \quad (\text{true})
\][/tex]
So, from the analysis, the integer [tex]\( 53 \)[/tex] makes the inequality false.
### Conclusion:
The integer that makes the inequality [tex]\( 4(n - 5) < 3(n + 11) \)[/tex] false is [tex]\( \boxed{53} \)[/tex].
