Question

simplify.\\(
\dfrac{3 + \dfrac{7}{4x}}{1 - \dfrac{1}{8x}}
\\)

Explanation:

Step1: Combine numerator terms

Rewrite 3 as $\frac{12x}{4x}$, then add:
$3+\frac{7}{4x} = \frac{12x}{4x}+\frac{7}{4x} = \frac{12x+7}{4x}$

Step2: Combine denominator terms

Rewrite 1 as $\frac{8x}{8x}$, then subtract:
$1-\frac{1}{8x} = \frac{8x}{8x}-\frac{1}{8x} = \frac{8x-1}{8x}$

Step3: Rewrite division as multiplication

Divide the two fractions by multiplying by the reciprocal:
$\frac{\frac{12x+7}{4x}}{\frac{8x-1}{8x}} = \frac{12x+7}{4x} \times \frac{8x}{8x-1}$

Step4: Simplify the expression

Cancel $4x$ and $8x$ (where $x
eq 0$):
$\frac{12x+7}{4x} \times \frac{8x}{8x-1} = (12x+7) \times \frac{2}{8x-1} = \frac{2(12x+7)}{8x-1}$

Step5: Expand the numerator

Distribute the 2 in the numerator:
$\frac{2(12x+7)}{8x-1} = \frac{24x+14}{8x-1}$

Answer:

$\frac{24x+14}{8x-1}$ (where $x
eq 0, x
eq \frac{1}{8}$)