Questions: Simplify the following complex rational expression. (x-4)/(x-(4/(x-3))) Select the correct choice below and fill in the answer box to complete your choice. (Simplify your answer.) A. (x-4)/(x-(4/(x-3)))= x neq 3,4 B. (x-4)/(x-(4/(x-3)))= x neq -4,-3,1 C. (x-4)/(x-(4/(x-3)))= , x neq -1,3,4

Simplify the following complex rational expression. (x-4)/(x-(4/(x-3))) Select the correct choice below and fill in the answer box to complete your choice. (Simplify your answer.) A. (x-4)/(x-(4/(x-3)))= x neq 3,4 B. (x-4)/(x-(4/(x-3)))= x neq -4,-3,1 C. (x-4)/(x-(4/(x-3)))= , x neq -1,3,4
Transcript text: Simplify the following complex rational expression. \[ \frac{x-4}{x-\frac{4}{x-3}} \] Select the correct choice below and fill in the answer box to complete your choice. (Simplify your answer.) A. $\frac{x-4}{x-\frac{4}{x-3}}=$ $\square$ $x \neq 3,4$ B. $\frac{x-4}{x-\frac{4}{x-3}}=$ $\square$ $x \neq-4,-3,1$ c. \[ \frac{x-4}{x-\frac{4}{x-3}}= \] $\square$ $, x \neq-1,3,4$
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Solution

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Solution Steps

Step 1: Rewrite the Expression

We start with the complex rational expression:

\[ \frac{x-4}{x-\frac{4}{x-3}} \]

To simplify this, we rewrite the denominator:

\[ x - \frac{4}{x-3} = \frac{x(x-3) - 4}{x-3} = \frac{x^2 - 3x - 4}{x-3} \]

Thus, the expression becomes:

\[ \frac{x-4}{\frac{x^2 - 3x - 4}{x-3}} = (x-4) \times \frac{x-3}{x^2 - 3x - 4} \]

Step 2: Factor the Numerator and Denominator

Next, we factor both the numerator and the denominator:

  • The numerator \(x - 4\) remains as is.
  • The denominator \(x^2 - 3x - 4\) factors to \((x - 4)(x + 1)\).

So we have:

\[ \frac{(x - 4)(x - 3)}{(x - 4)(x + 1)} \]

Step 3: Simplify the Expression

We can now simplify the expression by canceling the common factor \((x - 4)\):

\[ \frac{x - 3}{x + 1} \]

Final Answer

The simplified expression is:

\[ \frac{x - 3}{x + 1}, \quad x \neq 4 \]

Thus, the final answer is:

\(\boxed{\frac{x - 3}{x + 1}, \, x \neq 4}\)

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