Sure! Let's solve the inequality step-by-step.
Given inequality:
[tex]\[
\left|\frac{x-3}{5}\right| < 2
\][/tex]
The absolute value inequality [tex]\(\left| A \right| < B\)[/tex] can be split into two inequalities:
[tex]\[
- B < A < B
\][/tex]
So, we can rewrite the given inequality as:
[tex]\[
-2 < \frac{x-3}{5} < 2
\][/tex]
This compound inequality can be split into two individual inequalities:
1. [tex]\(-2 < \frac{x-3}{5}\)[/tex]
2. [tex]\(\frac{x-3}{5} < 2\)[/tex]
Now, let's solve each part individually.
### Solving [tex]\(-2 < \frac{x-3}{5}\)[/tex]:
1. Multiply all parts of the inequality by 5 to get rid of the fraction:
[tex]\[
-2 \times 5 < \frac{x-3}{5} \times 5 \implies -10 < x - 3
\][/tex]
2. Add 3 to all parts of the inequality:
[tex]\[
-10 + 3 < x - 3 + 3 \implies -7 < x
\][/tex]
Thus, one part of the inequality is:
[tex]\[
-7 < x
\][/tex]
### Solving [tex]\(\frac{x-3}{5} < 2\)[/tex]:
1. Multiply all parts of the inequality by 5 to get rid of the fraction:
[tex]\[
\frac{x-3}{5} \times 5 < 2 \times 5 \implies x - 3 < 10
\][/tex]
2. Add 3 to all parts of the inequality:
[tex]\[
x - 3 + 3 < 10 + 3 \implies x < 13
\][/tex]
Thus, the other part of the inequality is:
[tex]\[
x < 13
\][/tex]
### Combining the results:
We combine the two results to form the compound inequality:
[tex]\[
-7 < x < 13
\][/tex]
### Interval Notation:
The solution set in interval notation is:
[tex]\[
(-7, 13)
\][/tex]
So, the final solution to the given inequality [tex]\(\left|\frac{x-3}{5}\right| < 2\)[/tex] in interval notation is [tex]\((-7, 13)\)[/tex].
