For all real numbers x such that x \neq 0 and x \neq -1, which of the following expression… — ACT

For all real numbers $x$ such that $x \neq 0$ and $x \neq -1$ , which of the following expressions is equivalent to $\frac{3}{x} + \frac{2}{x + 1}$ ?

\frac{5x + 3}{x(x + 1)}

To add the rational expressions, find the common denominator

x(x + 1)

. Multiplying the first term by

\frac{x + 1}{x + 1}

and the second term by

\frac{x}{x}

gives

\frac{3(x + 1) + 2x}{x(x + 1)}

. Expanding the numerator yields

3x + 3 + 2x

, which simplifies to

5x + 3

. Thus, the equivalent expression is

\frac{5x + 3}{x(x + 1)}

Identify a common denominator for the two rational expressions.

The common denominator for the denominators

x

and

x + 1

x(x + 1)

To add fractions with different denominators, they must be written with a common denominator.

Rewrite each fraction with the common denominator by multiplying the numerator and denominator of the first term by

(x + 1)

and the second term by

x

\frac{3(x + 1)}{x(x + 1)} + \frac{2x}{x(x + 1)}

This scales the fractions to have matching denominators without changing their value.

Add the numerators together over the common denominator and simplify.

\frac{3x + 3 + 2x}{x(x + 1)} = \frac{5x + 3}{x(x + 1)}

Combining like terms simplifies the numerator to its final form.

Adding rational expressions by finding a common denominator

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