Question

Find the quotient.

$\frac{\mathbf{x}\mathbf{+}\mathbf{3}}{\mathbf{2}\mathbf{x}\mathbf{+}\mathbf{6}}\mathbf{÷}\frac{\mathbf{3}\mathbf{x}\mathbf{-}\mathbf{6}}{\mathbf{4}\mathbf{x}\mathbf{-}\mathbf{8}}$

Short answer

The final answer is $\frac{\mathbf{2}}{\mathbf{3}}$.

Step-by-step solution

Step 1. Define a rational function.

A function of the form $y=\frac{p}{q}$ is called a rational function, where $p$ and $q$ are the polynomials and $q\ne 0$.

Step 2. Define the quotient of two expressions ab and cd.

The quotient of two expressions $\frac{a}{b}$and $\frac{c}{d}$, where $b\ne 0$and $d\ne 0$, is given below.

$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}\times \frac{d}{c}=\frac{ac}{bd}\mathrm{...........}\left(1\right)$

Step 3. Calculate the quotient x+32x+6÷3x−64x−8.

Use (1) to find the quotient $\frac{x+3}{2x+6}÷\frac{3x-6}{4x-8}$.

$\begin{array}{c}\frac{x+3}{2x+6}÷\frac{3x-6}{4x-8}=\left(\frac{x+3}{2x+6}\right)\times \left(\frac{4x-8}{3x-6}\right)\\ =\frac{(x+3)(4x-8)}{(2x+6)(3x-6)}\\ =\frac{(x+3)4(x-2)}{2(x+3)3(x-2)}\\ =\frac{2}{3}\end{array}$

Therefore, the final answer is $\frac{2}{3}$.