Certainly! Let's solve the inequality [tex]\(-1 < \frac{x+5}{5} - 1 < 2\)[/tex] step by step.
1. Separate the Given Inequality into Two Inequalities:
The compound inequality can be broken down into two separate inequalities:
[tex]\[
-1 < \frac{x+5}{5} - 1
\][/tex]
and
[tex]\[
\frac{x+5}{5} - 1 < 2
\][/tex]
2. Solve the First Inequality: [tex]\(-1 < \frac{x+5}{5} - 1\)[/tex]
- First, isolate the term [tex]\(\frac{x+5}{5}\)[/tex] on the right-hand side by adding [tex]\(1\)[/tex] to both sides:
[tex]\[
-1 + 1 < \frac{x+5}{5} - 1 + 1
\][/tex]
[tex]\[
0 < \frac{x+5}{5}
\][/tex]
- Next, multiply both sides by [tex]\(5\)[/tex] to eliminate the denominator:
[tex]\[
0 \times 5 < \frac{x+5}{5} \times 5
\][/tex]
[tex]\[
0 < x + 5
\][/tex]
- Finally, subtract [tex]\(5\)[/tex] from both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
0 - 5 < x + 5 - 5
\][/tex]
[tex]\[
-5 < x
\][/tex]
3. Solve the Second Inequality: [tex]\(\frac{x+5}{5} - 1 < 2\)[/tex]
- First, isolate the term [tex]\(\frac{x+5}{5}\)[/tex] on the left-hand side by adding [tex]\(1\)[/tex] to both sides:
[tex]\[
\frac{x+5}{5} - 1 + 1 < 2 + 1
\][/tex]
[tex]\[
\frac{x+5}{5} < 3
\][/tex]
- Next, multiply both sides by [tex]\(5\)[/tex] to eliminate the denominator:
[tex]\[
\frac{x+5}{5} \times 5 < 3 \times 5
\][/tex]
[tex]\[
x + 5 < 15
\][/tex]
- Finally, subtract [tex]\(5\)[/tex] from both sides to solve for [tex]\(x\)[/tex]:
[tex]\[
x + 5 - 5 < 15 - 5
\][/tex]
[tex]\[
x < 10
\][/tex]
4. Combine the Results:
The two inequalities are:
[tex]\[
-5 < x \quad \text{and} \quad x < 10
\][/tex]
Combining these, we have:
[tex]\[
-5 < x < 10
\][/tex]
Therefore, the solution to the inequality [tex]\(-1 < \frac{x+5}{5} - 1 < 2\)[/tex] is:
[tex]\[
\boxed{-5 < x < 10}
\][/tex]
