Add two rational expressions whose denominators are opposites of each other:

$\frac{4u - 1}{3u - 1} + \frac{u}{1 - 3u}$

Step 1 — Recognize the opposite denominators. The denominators $3u - 1$ and $1 - 3u$ are opposites because $1 - 3u = -(3u - 1)$. To create a common denominator, multiply the numerator and denominator of the second fraction by $-1$:

$\frac{4u - 1}{3u - 1} + \frac{u \cdot (-1)}{(1 - 3u) \cdot (-1)}$

Step 2 — Simplify the second fraction. Multiplying gives $\frac{-u}{3u - 1}$. Now both fractions share the denominator $3u - 1$:

$\frac{4u - 1}{3u - 1} + \frac{-u}{3u - 1}$

Step 3 — Add the numerators over the common denominator:

$\frac{4u - 1 + (-u)}{3u - 1} = \frac{4u - 1 - u}{3u - 1}$

Step 4 — Combine like terms in the numerator. Group the $u$-terms: $4u - u = 3u$:

$\frac{3u - 1}{3u - 1}$

Step 5 — Simplify. The numerator and denominator are identical, so the expression equals $1$.

This example illustrates a case where adding rational expressions with opposite denominators produces a numerator that matches the denominator exactly, causing the entire expression to simplify to $1$.