To solve the rational equation:

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

First, we will find a common denominator for all the terms. The common denominator is $4x^2$. We will rewrite each term with this common denominator:

$\frac{7}{4x} = \frac{7 \cdot x}{4x \cdot x} = \frac{7x}{4x^2}$

$-\frac{3}{x^2} = \frac{-3 \cdot 4}{x^2 \cdot 4} = \frac{-12}{4x^2}$

$\frac{1}{x^2} = \frac{4}{4x^2}$

Substitute these back into the equation:

$\frac{7x}{4x^2} - \frac{12}{4x^2} = \frac{4}{4x^2}$

Combine all the terms over the common denominator:

$\frac{7x - 12}{4x^2} = \frac{4}{4x^2}$

Since the denominators are the same, we can equate the numerators:

$7x - 12 = 4$

Solve for $x$:

$7x = 4 + 12$

$7x = 16$

$x = \frac{16}{7}$

Therefore, the solution is:

$x = \frac{16}{7}$

Do you want more details or have any questions?

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1. How to find the common denominator of more complex rational expressions?
2. What happens if the equation has no solution?
3. How to handle rational equations with extraneous solutions?
4. What if the denominators contain polynomial expressions?
5. How to solve rational inequalities?

Tip: Always check the solution of a rational equation to ensure it doesn't make any denominator zero, which would make the expression undefined.