Question

finding the sum of rational expressions with common denominators
these rational expressions all have the same denominator. write the sum of their numerators over the common denominator.
\\(\frac{2x + 1}{x - 3} + \frac{x}{x - 3} + \frac{4}{x - 3}\\)
options:
\\(\frac{x + 5}{x - 3}\\)
\\(\frac{3x + 5}{x - 3}\\)
\\(\frac{8x^2 + 4x}{x - 3}\\)
\\(\frac{3x + 5}{3(x - 3)}\\)

Explanation:

Step1: Identify the rule for adding rational expressions with common denominators

When adding rational expressions with the same denominator, we use the rule: \(\frac{a}{c}+\frac{b}{c}+\frac{d}{c}=\frac{a + b + d}{c}\). Here, the common denominator is \(x - 3\), and the numerators are \(2x+1\), \(x\), and \(4\).

Step2: Add the numerators

Add the numerators together: \((2x + 1)+x + 4\). Combine like terms: \(2x+x=3x\) and \(1 + 4=5\). So the sum of the numerators is \(3x+5\).

Step3: Write the sum over the common denominator

Using the rule from Step 1, we write the sum of the numerators over the common denominator \(x - 3\). So the result is \(\frac{3x + 5}{x - 3}\).

Answer:

\(\frac{3x + 5}{x - 3}\) (corresponding to the option with this expression)