Question

which is equivalent to \\(\frac{2+\frac{3}{x}}{4-\frac{2}{3x}}\\)?
\\(\bigcirc\\) a. \\(\frac{1}{2}\\)
\\(\bigcirc\\) b. \\(\frac{2x+3}{4x-2}\\)
\\(\bigcirc\\) c. \\(\frac{6x+9}{12x-2}\\)
\\(\bigcirc\\) d. \\(\frac{6x+3}{12x-2}\\)
\\(\bigcirc\\) e. \\(\frac{3}{2}\\)

Explanation:

Step1: Simplify numerator

To simplify \(2 + \frac{3}{x}\), find a common denominator (which is \(x\)):
\(2+\frac{3}{x}=\frac{2x}{x}+\frac{3}{x}=\frac{2x + 3}{x}\)

Step2: Simplify denominator

To simplify \(4-\frac{2}{3x}\), find a common denominator (which is \(3x\)):
\(4-\frac{2}{3x}=\frac{12x}{3x}-\frac{2}{3x}=\frac{12x - 2}{3x}\)

Step3: Divide numerator by denominator

Now, the original expression \(\frac{2+\frac{3}{x}}{4-\frac{2}{3x}}\) becomes \(\frac{\frac{2x + 3}{x}}{\frac{12x - 2}{3x}}\). When dividing fractions, we multiply by the reciprocal:
\(\frac{2x + 3}{x}\times\frac{3x}{12x - 2}\)

Step4: Cancel common terms

The \(x\) in the numerator and denominator cancels out:
\(\frac{(2x + 3)\times3}{12x - 2}=\frac{6x + 9}{12x - 2}\)

Answer:

C. \(\frac{6x+9}{12x-2}\)