To simplify the given continued fraction, we will work from the innermost expression outwards.

Question

ScanSolve AI Tutor · Accepted Answer

To simplify the given continued fraction, we will work from the innermost expression outwards.

The expression is:

\frac{1}{2 - \frac{1}{2 - \frac{1}{2 - x}}}

Simplify the innermost denominator.
The innermost denominator is $2 - x$ . This cannot be simplified further.
Simplify the expression $2 - \frac{1}{2 - x}$ .
To combine these terms, we find a common denominator.
$2 - \frac{1}{2 - x} = \frac{2(2 - x)}{2 - x} - \frac{1}{2 - x} = \frac{4 - 2x - 1}{2 - x} = \frac{3 - 2x}{2 - x}$
Simplify the fraction $\frac{1}{2 - \frac{1}{2 - x}}$ .
We use the result from Step 2.
$\frac{1}{2 - \frac{1}{2 - x}} = \frac{1}{\frac{3 - 2x}{2 - x}}$
To divide by a fraction, we multiply by its reciprocal.
$\frac{1}{\frac{3 - 2x}{2 - x}} = \frac{2 - x}{3 - 2x}$
Simplify the expression $2 - \frac{1}{2 - \frac{1}{2 - x}}$ .
We use the result from Step 3.
$2 - \frac{2 - x}{3 - 2x}$
Again, find a common denominator.
$\frac{2(3 - 2x)}{3 - 2x} - \frac{2 - x}{3 - 2x} = \frac{6 - 4x - (2 - x)}{3 - 2x} = \frac{6 - 4x - 2 + x}{3 - 2x} = \frac{4 - 3x}{3 - 2x}$
Simplify the entire expression $\frac{1}{2 - \frac{1}{2 - \frac{1}{2 - x}}}$ .
We use the result from Step 4.
$\frac{1}{\frac{4 - 3x}{3 - 2x}}$
Multiply by the reciprocal.
$\frac{1}{\frac{4 - 3x}{3 - 2x}} = \frac{3 - 2x}{4 - 3x}$
The simplified expression is:
$\frac{3 - 2x}{4 - 3x}$

Answer