1Cademy - Simplifying $$\frac{2u}{u-1} + \frac{1}{u}

Learn Before

How to Add or Subtract Rational Expressions

Example

Simplifying $\frac{2u}{u-1} + \frac{1}{u} - \frac{2u-1}{u^2-u}$

Simplify a combination of three rational expressions with unlike denominators, where one denominator is the product of the other two after GCF factoring:

$\frac{2u}{u-1} + \frac{1}{u} - \frac{2u-1}{u^2-u}$

Step 1 — Factor the denominators and find the LCD. The first two denominators, $(u - 1)$ and $u$ , are already fully factored. The third denominator $u^2 - u$ has a GCF of $u$ : $u^2 - u = u(u - 1)$ . Since $u(u - 1)$ already contains both of the other denominators as factors, the LCD is $u(u - 1)$ .

Step 2 — Rewrite each fraction with the LCD. The first fraction $\frac{2u}{u - 1}$ is missing the factor $u$ ; multiply its numerator and denominator by $u$ . The second fraction $\frac{1}{u}$ is missing the factor $(u - 1)$ ; multiply its numerator and denominator by $(u - 1)$ . The third fraction already has the LCD:

$\frac{2u \cdot u}{(u - 1) \cdot u} + \frac{1 \cdot (u - 1)}{u \cdot (u - 1)} - \frac{2u - 1}{u(u - 1)}$

Simplify each numerator: $2u \cdot u = 2u^2$ and $1 \cdot (u - 1) = u - 1$ :

$\frac{2u^2}{u(u - 1)} + \frac{u - 1}{u(u - 1)} - \frac{2u - 1}{u(u - 1)}$

Step 3 — Combine into one rational expression. Write a single fraction with the combined numerators over the common denominator:

$\frac{2u^2 + (u - 1) - (2u - 1)}{u(u - 1)}$

Distribute the negative sign to the third numerator: $-(2u - 1) = -2u + 1$ . Combine all terms:

$\frac{2u^2 + u - 1 - 2u + 1}{u(u - 1)} = \frac{2u^2 - u}{u(u - 1)}$

Step 4 — Factor the numerator and simplify. Extract the GCF from the numerator: $2u^2 - u = u(2u - 1)$ . Cancel the common factor $u$ :

$\frac{u(2u - 1)}{u(u - 1)} = \frac{2u - 1}{u - 1}$

This example extends the standard addition and subtraction procedure to three rational expressions at once. A key observation is that factoring the third denominator $u^2 - u = u(u - 1)$ reveals it to be the product of the other two denominators, so the LCD equals the third denominator itself. After combining all three numerators — being careful to distribute the subtraction sign — the combined numerator factors and shares the factor $u$ with the denominator, allowing simplification.

Updated 2026-04-21

Contributors are:

Who are from:

References

OpenStax Elementary Algebra

Learn Before

Related

Learn After