Answer :
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].
[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].
