Question: 3 — Adding and Subtracting Rational Expressions
LEARNING OBJECTIVE: Add or subtract two rational expressions with an uncommon denominator.
The rational expression
$\frac{5}{2x} + \frac{3x}{x + 1}$ is equivalent to ________.
1. **State the problem:**
We want to add the rational expressions $\frac{5}{2x}$ and $\frac{3x}{x + 1}$ which have different denominators.
2. **Find a common denominator:**
The denominators are $2x$ and $x + 1$. The least common denominator (LCD) is $$2x(x + 1)$$.
3. **Rewrite each fraction with the LCD:**
$$\frac{5}{2x} = \frac{5(x + 1)}{2x(x + 1)} = \frac{5x + 5}{2x(x + 1)}$$
$$\frac{3x}{x + 1} = \frac{3x(2x)}{2x(x + 1)} = \frac{6x^2}{2x(x + 1)}$$
4. **Add the numerators over the common denominator:**
$$\frac{5x + 5}{2x(x + 1)} + \frac{6x^2}{2x(x + 1)} = \frac{5x + 5 + 6x^2}{2x(x + 1)}$$
5. **Combine like terms in the numerator:**
$$\frac{6x^2 + 5x + 5}{2x(x + 1)}$$
6. **Check if numerator can be factored or simplified:**
The quadratic $6x^2 + 5x + 5$ does not factor nicely with integer factors, so this is the simplified form.
7. **Final answer:**
$$\frac{6x^2 + 5x + 5}{2x(x + 1)}$$
This matches option (b).