Question

Solve for x


`\ \textless \ br /\ \textgreater \ x = (1)/(2-(1)/(2-(1)/(2-x) ) ) \: \: \: , (x \\eq 2)`
Help me with this!​

Answer 1

`x= (3-2x)/(4-3x)`

Explanation:

Given:

`x=(1)/(2-(1)/(2-(1)/(2-x)))}`

Simplify:

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

Therefore:

`x=(1)/(2-(1)/((3-2x)/(2-x)))`

Simplify:

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

Therefore:

`x & =(1)/((4-3x)/(3-2x))`

`\textsf{Apply the fraction rule}: \quad (1)/((b)/(c))=(c)/(b)`

`\implies x& = (3-2x)/(4-3x)`