Skill: Add rational expressions with common denominators

When rational expressions have the same denominator, keep that denominator and add the numerators:

$$\frac{A(x)}{B(x)}+\frac{C(x)}{B(x)} =\frac{A(x)+C(x)}{B(x)}.$$

The denominator tells you the size of each algebraic unit, so it is not added. First note any restriction: the denominator cannot equal zero.

Example:

$$\frac{2x+3}{x-4}+\frac{x-7}{x-4}$$

Step 1: State the restriction.

The denominator is $x-4$, so

$$x-4\ne 0 \quad\Rightarrow\quad x\ne 4.$$

Step 2: Keep the common denominator.

Since both expressions have denominator $x-4$, write one fraction with that denominator:

$$\frac{(2x+3)+(x-7)}{x-4}.$$

Step 3: Combine like terms in the numerator.

$$(2x+3)+(x-7) =2x+x+3-7 =3x-4.$$

Therefore,

$$\frac{2x+3}{x-4}+\frac{x-7}{x-4} =\frac{3x-4}{x-4}, \qquad x\ne 4.$$

Do not add the denominators. For example, the denominator is not $(x-4)+(x-4)$. Also, do not cancel terms across addition; simplify only after combining the numerators, and keep the restriction $x\ne4$.