In Exercises 33–68, add or subtract as indicated. 3/(x+1) − 3/x

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Identify the two rational expressions to be subtracted: $\frac{3}{x+1} - \frac{3}{x}$.

Find the least common denominator (LCD) of the two fractions. Since the denominators are $x+1$ and $x$, the LCD is $x(x+1)$.

Rewrite each fraction with the LCD as the new denominator by multiplying numerator and denominator appropriately: $\frac{3}{x+1} = \frac{3x}{x(x+1)}$ and $\frac{3}{x} = \frac{3(x+1)}{x(x+1)}$.

Subtract the numerators over the common denominator: $\frac{3x}{x(x+1)} - \frac{3(x+1)}{x(x+1)} = \frac{3x - 3(x+1)}{x(x+1)}$.

Simplify the numerator by distributing and combining like terms: $3x - 3(x+1) = 3x - 3x - 3 = -3$, so the expression becomes $\frac{-3}{x(x+1)}$.

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Adding and Subtracting Rational Expressions

To add or subtract rational expressions, they must have a common denominator. This involves rewriting each fraction so that their denominators are the same, allowing the numerators to be combined directly.

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Finding the Least Common Denominator (LCD)

The LCD is the smallest expression that both denominators divide into evenly. For expressions like 3/(x+1) and 3/x, the LCD is the product of the distinct factors, here x(x+1), which allows combining the fractions.

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Simplifying Rational Expressions

After combining the numerators over the common denominator, simplify the resulting expression by factoring and reducing common factors. This step ensures the final answer is in simplest form.

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