To solve the inequality

\[
\frac{3}{x} + 1 < \frac{1}{x}
\]

we can proceed as follows:

1. **Combine Terms on the Left Side**: Start by moving \(\frac{1}{x}\) from the right side to the left:
   \[
   \frac{3}{x} + 1 - \frac{1}{x} < 0
   \]

2. **Simplify the Left Side**: Combine \(\frac{3}{x}\) and \(-\frac{1}{x}\) by combining the numerators:
   \[
   \frac{3 - 1}{x} + 1 < 0
   \]
   \[
   \frac{2}{x} + 1 < 0
   \]

3. **Isolate the Fraction**: Now, move the 1 to the other side:
   \[
   \frac{2}{x} < -1
   \]

4. **Clear the Denominator**: Multiply both sides by \(x\), keeping in mind that \(x\) cannot be zero, and the sign of \(x\) affects the inequality:
   - If \(x > 0\), multiplying by \(x\) keeps the inequality direction the same:
     \[
     2 < -x
     \]
     Divide by \(-1\) (which reverses the inequality):
     \[
     x < -2
     \]

   - If \(x < 0\), multiplying by \(x\) reverses the inequality:
     \[
     2 > -x
     \]
     Divide by \(-1\) again:
     \[
     x > -2
     \]

5. **Combine Results**: For the inequality to hold, \(x\) must satisfy:
   \[
   x < -2
   \]

### Solution
The solution to the inequality is:
\[
x < -2
\]

Let me know if you'd like more details or have any questions!

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**Further Questions:**
1. How would the solution change if the inequality were \(\frac{3}{x} + 1 > \frac{1}{x}\)?
2. What if the inequality were an equation, such as \(\frac{3}{x} + 1 = \frac{1}{x}\)?
3. How does multiplying by a negative affect inequality solutions?
4. Can you solve a similar inequality like \(\frac{5}{x} + 2 < \frac{3}{x}\)?
5. How would you graph the solution to this inequality?

**Tip:** Remember that multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign.

Explore the solution to the inequality 3/x + 1 < 1/x with a comprehensive, step-by-step guide. This explanation includes key algebraic concepts such as isolating fractions and understanding the effects of multiplying by negative numbers, suitable for students in Grades 8-10.

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