Question

Solve for x
$\ \textless \ br /\ \textgreater \ x = \dfrac{1}{2-\dfrac{1}{2-\dfrac{1}{2-x} } } \: \: \: , (x \neq 2)$
Help me with this!​

Answer 1

Answer:

$x= \dfrac{3-2x}{4-3x}$

Step-by-step explanation:

Given:

$x=\dfrac{1}{2-\dfrac{1}{2-\dfrac{1}{2-x}}}}$

Simplify:

$\begin{aligned}2-\dfrac{1}{2-x} & =\dfrac{2(2-x)-1}{2-x}\\\\ & =\dfrac{3-2x}{2-x}\end{alilgned}$

Therefore:

$x=\dfrac{1}{2-\dfrac{1}{\dfrac{3-2x}{2-x}}}$

Simplify:

$\begin{aligned}2-\dfrac{1}{\dfrac{3-2x}{2-x}} & =2-\dfrac{2-x}{3-2x}\\\\ & =\dfrac{2(3-2x)-(2-x)}{3-2x}\\\\ & = \dfrac{6-4x-2+x}{3-2x}\\\\ & = \dfrac{4-3x}{3-2x}\end{aligned}$

Therefore:

$x & =\dfrac{1}{\dfrac{4-3x}{3-2x}}$

$\textsf{Apply the fraction rule}: \quad \dfrac{1}{\frac{b}{c}}=\dfrac{c}{b}$

$\implies x& = \dfrac{3-2x}{4-3x}$