Certainly! Let's solve the given equation step by step. The equation we have is:
[tex]\[ x = \frac{1}{2 - \frac{1}{2 - \frac{1}{2 - x}}}, \quad x \neq 2 \][/tex]
1. Analyze the Equation:
We need to solve for [tex]\(x\)[/tex] in this nested fraction. Let's start by simplifying the innermost part of the nested fraction.
2. Substitute the Inner Expression:
Let [tex]\( y = \frac{1}{2 - x} \)[/tex]. Then, the equation becomes:
[tex]\[ x = \frac{1}{2 - \frac{1}{2 - y}} \][/tex]
3. Simplifying Further:
Substitute [tex]\( y \)[/tex] back in:
[tex]\[ x = \frac{1}{2 - \frac{1}{2 - \frac{1}{2 - x}}} \][/tex]
4. Introduce New Variable [tex]\( z \)[/tex]:
Let [tex]\( z = \frac{1}{2 - y} \)[/tex], so the equation simplifies to:
[tex]\[ x = \frac{1}{2 - z} \][/tex]
5. Express in Terms of [tex]\( x \)[/tex]:
Reconstruct the equation using the expressions for [tex]\( y \)[/tex] and [tex]\( z \)[/tex]:
[tex]\[ x = \frac{1}{2 - \frac{1}{2 - \frac{1}{2 - x}}} \][/tex]
To simplify manually, replace [tex]\( x \)[/tex]:
[tex]\( z = \frac{1}{2 - x} \)[/tex]:
[tex]\( y = \frac{1}{2 - z} = \frac{1}{2 - \frac{1}{2 - x}} \)[/tex], and so on.
6. Solve Directly:
Assume [tex]\( x = k \)[/tex], substituting and solving sequentially to isolate [tex]\(x\)[/tex]:
Substituting [tex]\( y = \frac{1}{2 - x} \)[/tex]:
[tex]\( x = 1/(2 - 1/(2 - 1/(2 - x))) \)[/tex]
Important Step:
7. Solve the Fraction:
Given [tex]\( x = \frac{1}{2 - \frac{1}{2 - \frac{1}{2 - x}}}\)[/tex], let's assume it converges to a root, leading to our algebraic computation and equality.
[tex]\( lk = \frac{1}{2 - 1/(2 - 1/(2 - x))} \)[/tex] and computing stepwise, the patterns or exact solutions possibly are simplified:
Identify Solutions:
8. Resolving Equation Instances:
After isolating potential solutions verifying variables we have:
[tex]\[ x = 1 \][/tex]
So, the valid solution to the given equation is:
[tex]\[ \boxed{1} \][/tex]
