Let's solve the given problems step-by-step.
### Problem 1: Solving the Inequality [tex]\(3x + 1 < 10\)[/tex]
1. Step 1: Isolate the term involving [tex]\(x\)[/tex].
We start with the inequality:
[tex]\[
3x + 1 < 10
\][/tex]
To isolate [tex]\(3x\)[/tex], we subtract 1 from both sides of the inequality:
[tex]\[
3x < 10 - 1
\][/tex]
Simplifying the right side, we get:
[tex]\[
3x < 9
\][/tex]
2. Step 2: Solve for [tex]\(x\)[/tex].
Now, we divide both sides of the inequality by 3 to solve for [tex]\(x\)[/tex]:
[tex]\[
x < \frac{9}{3}
\][/tex]
Simplifying the fraction on the right side, we obtain:
[tex]\[
x < 3
\][/tex]
3. Step 3: Consider the domain of [tex]\(x\)[/tex].
Since we're dealing with a linear inequality, there are no restrictions on [tex]\(x\)[/tex] other than it being less than 3. Therefore, [tex]\(x\)[/tex] can be any real number less than 3.
4. Step 4: Express the solution.
The solution to the inequality is:
[tex]\[
x \in (-\infty, 3)
\][/tex]
In interval notation, this can be written as:
[tex]\[
(-\infty < x < 3)
\][/tex]
So, the solution to the inequality [tex]\(3x + 1 < 10\)[/tex] is:
[tex]\[
(-\infty < x < 3)
\][/tex]
### Problem 2: Simplifying the Fraction [tex]\(\frac{x}{3}\)[/tex]
1. Step 1: Identify the fraction to simplify.
We start with the given fraction:
[tex]\[
\frac{x}{3}
\][/tex]
2. Step 2: Simplify the fraction.
In this case, the fraction [tex]\(\frac{x}{3}\)[/tex] is already in its simplest form because [tex]\(x\)[/tex] and 3 have no common factors other than 1.
3. Step 3: Verify the simplified form.
Since there's nothing more to reduce or simplify, the fraction stays as:
[tex]\[
\frac{x}{3}
\][/tex]
So, the simplified form of the fraction [tex]\(\frac{x}{3}\)[/tex] is:
[tex]\[
\frac{x}{3}
\][/tex]
### Conclusion
To summarize:
1. The solution to the inequality [tex]\(3x + 1 < 10\)[/tex] is:
[tex]\[
(-\infty < x < 3)
\][/tex]
2. The simplified form of the fraction [tex]\(\frac{x}{3}\)[/tex] is:
[tex]\[
\frac{x}{3}
\][/tex]
These results give us a comprehensive understanding of both the inequality and the fraction simplification.
