To solve the equation [tex]\( x = \frac{1}{2 - \frac{1}{2 - \frac{1}{2 - x}}} \)[/tex], let's proceed with a step-by-step approach:
1. Start with the given equation:
[tex]\[
x = \frac{1}{2 - \frac{1}{2 - \frac{1}{2 - x}}}
\][/tex]
2. Substitute [tex]\( y \)[/tex] to simplify the nested fraction:
Let [tex]\( y = \frac{1}{2 - \frac{1}{2 - x}} \)[/tex]. Then the equation becomes:
[tex]\[
x = \frac{1}{2 - y}
\][/tex]
3. Express [tex]\( y \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
y = \frac{1}{2 - \frac{1}{2 - x}}
\][/tex]
4. Let's solve the inner fraction:
[tex]\[
y = \frac{1}{2 - \frac{1}{2 - x}}
\][/tex]
5. Express the inner denominator:
Solve the inner [tex]\(2 - x\)[/tex]:
[tex]\[
z = 2 - x
\][/tex]
The inner fraction becomes:
[tex]\[
y = \frac{1}{2 - \frac{1}{z}}
\][/tex]
6. Solve for [tex]\( y \)[/tex]:
Substitute back [tex]\( z \)[/tex] and simplify:
[tex]\[
y = \frac{1}{2 - \frac{1}{2 - x}}
\][/tex]
[tex]\[
y = \frac{1}{\frac{2(2 - x) - 1}{2 - x}}
\][/tex]
[tex]\[
y = \frac{2 - x}{4 - x - 1}
\][/tex]
[tex]\[
y = \frac{2 - x}{3 - x}
\][/tex]
7. Thus, we have two equations:
[tex]\[
x = \frac{1}{2 - y}
\][/tex]
And
[tex]\[
y = \frac{2 - x}{3 - x}
\][/tex]
8. Since both expressions equal [tex]\( y \)[/tex], set them equal to solve for [tex]\( x \)[/tex]:
[tex]\[
\frac{1}{2 - y} = \frac{2 - x}{3 - x}
\][/tex]
9. Solve for [tex]\( y \)[/tex]:
Cross-multiplying gives:
[tex]\[
y(3 - x) = 2 - x
\][/tex]
Distribute [tex]\( y \)[/tex]:
[tex]\[
3y - xy = 2 - x
\][/tex]
10. Isolate [tex]\( y \)[/tex]:
Since [tex]\( y = 2 - x \)[/tex] and [tex]\( y = 1 \)[/tex]:
[tex]\[
y = 1
\][/tex]
Hence, the solution to the equation [tex]\( x = \frac{1}{2 - \frac{1}{2 - \frac{1}{2 - x}}} \)[/tex] is [tex]\(\boxed{1}\)[/tex].
