Question
The rational expression \( \frac{2 x+3}{x+4}+\frac{8}{x} \) is equivalent to a.) \( \frac{2 x^{2}+11 x+32}{x(x+4)} \) b.) \( \frac{2 x+11}{2 x+4} \) c.) \( \frac{2 x^{2}-5 x-32}{x(x+4)} \) d.) \( \frac{2 x-11}{2 x-4} \)
Ask by Stanley Gonzalez. in Germany
Dec 21,2024
Real Tutor Solution
Tutor-Verified Answer
Answer
The equivalent expression is \( \frac{2x^{2}+11x+32}{x(x+4)} \).
Solution
To determine which expression is equivalent to \( \frac{2x + 3}{x + 4} + \frac{8}{x} \), let's perform the addition step by step.
**Step 1: Find a Common Denominator**
The denominators are \( x + 4 \) and \( x \). The least common denominator (LCD) is \( x(x + 4) \).
**Step 2: Rewrite Each Fraction with the LCD**
\[
\frac{2x + 3}{x + 4} = \frac{(2x + 3) \cdot x}{x(x + 4)} = \frac{2x^2 + 3x}{x(x + 4)}
\]
\[
\frac{8}{x} = \frac{8 \cdot (x + 4)}{x(x + 4)} = \frac{8x + 32}{x(x + 4)}
\]
**Step 3: Add the Two Fractions**
\[
\frac{2x^2 + 3x}{x(x + 4)} + \frac{8x + 32}{x(x + 4)} = \frac{2x^2 + 3x + 8x + 32}{x(x + 4)} = \frac{2x^2 + 11x + 32}{x(x + 4)}
\]
**Conclusion:**
The simplified form of the given expression matches option **a.**
**Answer:** a.) \( \frac{2 x^{2}+11 x+32}{x(x+4)} \)
Reviewed and approved by the UpStudy tutoring team
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Explain 
Simplify this solution Mind Expander
To combine the rational expressions \( \frac{2x+3}{x+4} \) and \( \frac{8}{x} \), we'll find a common denominator and simplify. The common denominator between \( x+4 \) and \( x \) is \( x(x+4) \). Rewriting the expressions, we have: \[ \frac{2x+3}{x+4} = \frac{(2x+3)x}{x(x+4)} = \frac{2x^2 + 3x}{x(x+4)} \] And for the second term: \[ \frac{8}{x} = \frac{8(x+4)}{x(x+4)} = \frac{8x + 32}{x(x+4)} \] Now, we can combine the two fractions: \[ \frac{2x^2 + 3x + 8x + 32}{x(x+4)} = \frac{2x^2 + 11x + 32}{x(x+4)} \] Thus, the equivalent expression matches option (a): a.) \( \frac{2 x^{2}+11 x+32}{x(x+4)} \) So the answer is (a).
