Solving the Algebraic Equation for $(2x)^2$

We are given the equation:

$11 = \frac{11x}{1 - x}$

Our goal is to find the value of $(2x)^2$. First, let's solve for $x$.

Finding the Value of $x$

1. Divide both sides of the equation by 11:

   $ \frac{11}{11} = \frac{11x}{11(1 - x)} $

   $ 1 = \frac{x}{1 - x} $

2. Multiply both sides by $(1 - x)$ to eliminate the denominator:

   $ 1 \cdot (1 - x) = x $

   $ 1 - x = x $

3. Add $x$ to both sides to group the $x$ terms:

   $ 1 = x + x $

   $ 1 = 2x $

4. Divide by 2 to find the value of $x$:

   $ x = \frac{1}{2} $

Calculating $(2x)^2$

Now that we have found $x = \frac{1}{2}$, we can substitute this value into the expression $(2x)^2$:

$ (2x)^2 = \left(2 \cdot \frac{1}{2}\right)^2 $

First, calculate the term inside the parenthesis:

$ 2 \cdot \frac{1}{2} = 1 $

Now, square the result:

$ (1)^2 = 1 $

Final Result

Therefore, the value of $(2x)^2$ is 1.