Let's solve the equation step-by-step to determine if it has any solutions:
The equation given is:
[tex]\[
\frac{x}{4x - 16} - 2 = \frac{1}{x - 4}
\][/tex]
First, let's simplify and solve the equation.
Step 1: Notice that the denominator [tex]\(4x - 16\)[/tex] can be factored:
[tex]\[
4x - 16 = 4(x - 4)
\][/tex]
So, the equation becomes:
[tex]\[
\frac{x}{4(x - 4)} - 2 = \frac{1}{x - 4}
\][/tex]
Step 2: To combine the left side over a common denominator, we need both terms to have the same denominator. The common denominator is [tex]\(4(x - 4)\)[/tex]:
[tex]\[
\frac{x - 2 \cdot 4(x - 4)}{4(x - 4)} = \frac{1}{x - 4}
\][/tex]
Simplifying the numerator:
[tex]\[
x - 8(x - 4) = x - 8x + 32 = -7x + 32
\][/tex]
So, the equation now is:
[tex]\[
\frac{-7x + 32}{4(x - 4)} = \frac{1}{x - 4}
\][/tex]
Step 3: Since the denominators are equal, we can set the numerators equal to each other:
[tex]\[
-7x + 32 = 4
\][/tex]
Step 4: Solve for [tex]\(x\)[/tex]:
[tex]\[
-7x + 32 = 4
\][/tex]
Subtract 32 from both sides:
[tex]\[
-7x = 4 - 32
\][/tex]
[tex]\[
-7x = -28
\][/tex]
Divide by -7:
[tex]\[
x = 4
\][/tex]
Step 5: Check if [tex]\(x = 4\)[/tex] is a valid solution by substituting in the original equation:
Substitute [tex]\(x = 4\)[/tex] in [tex]\(4x - 16\)[/tex]:
[tex]\[
4(4) - 16 = 0
\][/tex]
This makes the denominator zero and thus the equation undefined for [tex]\(x = 4\)[/tex].
Therefore, [tex]\(x = 4\)[/tex] is not a valid solution because it leads to division by zero.
Since our earlier steps show [tex]\(x = 4\)[/tex] as the only solution, but it is invalid, we conclude that there are no valid solutions to the equation.
The answer is:
[tex]\[
\boxed{\text{No}}
\][/tex]
