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.