Skill: Subtract rational expressions

To subtract rational expressions:

1. State the values that are not allowed by the original denominators.
2. Factor the denominators.
3. Find the least common denominator (LCD).
4. Rewrite each fraction using the LCD.
5. Subtract the numerators, distributing the minus sign.
6. Simplify, while keeping the original restrictions.

Example

Simplify

$$\frac{2x}{x^2-9}-\frac{1}{x-3}.$$

Step 1: Find the restrictions.

The denominators cannot equal zero:

$$x^2-9\ne 0 \quad\text{and}\quad x-3\ne 0.$$

Factor the first denominator:

$$x^2-9=(x-3)(x+3).$$

Therefore,

$$x\ne 3 \quad\text{and}\quad x\ne -3.$$

These restrictions must remain in the final answer.

Step 2: Find the LCD.

The denominators are $(x-3)(x+3)$ and $x-3$, so the LCD is

$$(x-3)(x+3).$$

Step 3: Rewrite both fractions using the LCD.

The first fraction already has the LCD:

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

For the second fraction, multiply by $\frac{x+3}{x+3}$:

$$\frac{1}{x-3}\cdot\frac{x+3}{x+3} = \frac{x+3}{(x-3)(x+3)}.$$

Now subtract:

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

Step 4: Subtract the numerators.

Keep the common denominator and distribute the subtraction:

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

Simplify the numerator:

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

So,

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

Step 5: Simplify.

Cancel the common factor $x-3$:

$$\boxed{\frac{1}{x+3}}, \qquad x\ne 3,-3.$$

Even though $x-3$ was canceled, $x=3$ is still excluded because it made an original denominator equal to zero. Never subtract denominators; only the numerators are subtracted after the fractions have a common denominator.