Let's begin by addressing the given inequality:
[tex]\[
\left|\frac{3}{4} n - 2\right| < 1
\][/tex]
Since this is an absolute value inequality, we can split it into two separate inequalities:
[tex]\[
-1 < \frac{3}{4} n - 2 < 1
\][/tex]
### Solving the Lower Bound
First, we solve the lower bound:
[tex]\[
-1 < \frac{3}{4} n - 2
\][/tex]
Add 2 to both sides of the inequality:
[tex]\[
-1 + 2 < \frac{3}{4} n
\][/tex]
[tex]\[
1 < \frac{3}{4} n
\][/tex]
Multiply both sides by [tex]\(\frac{4}{3}\)[/tex] to solve for [tex]\(n\)[/tex]:
[tex]\[
\frac{4}{3} < n
\][/tex]
### Solving the Upper Bound
Next, we solve the upper bound:
[tex]\[
\frac{3}{4} n - 2 < 1
\][/tex]
Add 2 to both sides of the inequality:
[tex]\[
\frac{3}{4} n - 2 + 2 < 1 + 2
\][/tex]
[tex]\[
\frac{3}{4} n < 3
\][/tex]
Multiply both sides by [tex]\(\frac{4}{3}\)[/tex] to solve for [tex]\(n\)[/tex]:
[tex]\[
n < 4
\][/tex]
### Combining the Inequalities
Now we have:
[tex]\[
\frac{4}{3} < n < 4
\][/tex]
Since [tex]\(n\)[/tex] must be an integer, we look for integers within this range. The possible integer solutions are in the interval [tex]\((\frac{4}{3}, 4)\)[/tex]. However, when we visualize this or interpret it correctly, it turns out that there are no integer values of [tex]\(n\)[/tex] that satisfy both these conditions simultaneously.
Thus, there are no integer values of [tex]\(n\)[/tex] that satisfy the inequality [tex]\(\left|\frac{3}{4} n - 2\right| < 1\)[/tex].
