We are given the equation:

$\frac{7}{4x} - \frac{3}{x^2} = \frac{1}{2x^2}$

Step 1: Find the common denominator

The denominators are $4x$, $x^2$, and $2x^2$. To simplify the equation, the least common denominator (LCD) is $4x^2$. We'll multiply both sides of the equation by $4x^2$ to eliminate the fractions.

Multiplying each term by $4x^2$, we get:

$\left( \frac{7}{4x} \right) \cdot 4x^2 - \left( \frac{3}{x^2} \right) \cdot 4x^2 = \left( \frac{1}{2x^2} \right) \cdot 4x^2$

Step 2: Simplify each term

• First term: $\frac{7}{4x} \cdot 4x^2 = 7x$
• Second term: $\frac{3}{x^2} \cdot 4x^2 = 12$
• Third term: $\frac{1}{2x^2} \cdot 4x^2 = 2$

Now, the equation becomes:

$7x - 12 = 2$

Step 3: Solve for $x$

Add 12 to both sides:

$7x = 14$

Divide by 7:

$x = 2$

Final Answer:

The solution is $x = 2$.

Would you like more details, or do you have any questions?

Here are five related questions:

1. How do you find the least common denominator in similar equations?
2. Can you solve a similar equation where the constants are different?
3. How do you handle equations involving higher powers of $x$?
4. What are other methods to solve equations with fractions?
5. What if the solution involves negative values of $x$?

Tip: Always check your solution by substituting back into the original equation to verify.