The inequality in the image is:

$\frac{1}{x} > 2$

To solve this inequality for $x$, we proceed as follows:

Step 1: Isolate $x$

Multiply both sides of the inequality by $x$ to eliminate the fraction. However, since $x$ could be positive or negative, we must consider two cases.

Case 1: $x > 0$

• If $x > 0$, then multiplying by $x$ doesn't change the inequality sign: $1 > 2x \implies x < \frac{1}{2}$

Case 2: $x < 0$

• If $x < 0$, then multiplying by $x$ will reverse the inequality sign: $1 < 2x \implies x > \frac{1}{2}$ However, this would contradict our initial assumption that $x < 0$. Thus, there is no valid solution in this case.

Step 2: Combine the results

From Case 1, the solution is $x < \frac{1}{2}$, provided $x > 0$.

Final Solution

The solution to the inequality is: $0 < x < \frac{1}{2}$

This means $x$ must be positive and less than $\frac{1}{2}$.

Would you like any additional details or have any further questions?

Follow-up Questions:

1. What is the solution to $\frac{1}{x} < -2$?
2. How does the solution change if the inequality is $\frac{1}{x} \geq 2$?
3. What happens if $\frac{1}{x} > -2$?
4. Can you solve $\frac{1}{x} > \frac{1}{y}$ where $x$ and $y$ are both positive?
5. How do you solve a quadratic inequality such as $x^2 - 4x + 3 < 0$?

Tip:

When solving inequalities involving fractions, always consider the sign of the variable to correctly determine the direction of the inequality.