Adding fractions with unlike denominators

Fractions can only be combined when they share a denominator, because the denominator sets the size of each piece. When the denominators differ, the first job is always to rewrite every fraction over a common denominator.

For exactly two fractions, the quickest reliable method is cross-multiplication: $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$. The new denominator is the product $bd$, and the new numerator is $ad + bc$. Using the least common denominator instead keeps the numbers smaller, but $bd$ always works.

This is identical for algebraic fractions, since the letters behave just like numbers. To combine $\frac{1}{x} + \frac{1}{y}$ the common denominator is $xy$, giving $\frac{y}{xy} + \frac{x}{xy} = \frac{x + y}{xy}$.

A very common move is combining a whole term with a fraction. Write the whole term over the same denominator first: $2x = \frac{2x^2}{x}$, so $2x - \frac{8}{x} = \frac{2x^2 - 8}{x}$. The same trick turns $1 + \frac{2}{x}$ into $\frac{x + 2}{x}$.

Combine $\frac{2}{5} + \frac{3}{7}$. Cross-multiply: the numerator is $2 \times 7 + 5 \times 3 = 14 + 15 = 29$ and the denominator is $5 \times 7 = 35$. So the sum is $\frac{29}{35}$.

Write $3x + \frac{12}{x}$ as a single fraction. Rewrite $3x$ over $x$: $3x = \frac{3x^2}{x}$. Then $\frac{3x^2}{x} + \frac{12}{x} = \frac{3x^2 + 12}{x}$.

Combine $\frac{1}{x} + \frac{1}{2x}$. The common denominator is $2x$: $\frac{2}{2x} + \frac{1}{2x} = \frac{3}{2x}$.