Rationale of the Expression: 3x/x-4 - x+8/x-4
In this article, we will discuss the rationalization of the expression 3x/x-4 - x+8/x-4. We will go through the steps to simplify this expression and provide a detailed explanation of each step.
Step 1: Simplify the First Term
The first term of the expression is 3x/x-4. To simplify this term, we can start by dividing the numerator by the denominator.
$\frac{3x}{x-4} = 3\left(\frac{x}{x-4}\right)$
Now, we can see that the x in the numerator can be cancelled out with the x in the denominator, leaving us with:
$3\left(\frac{1}{1-\frac{4}{x}}\right)$
Step 2: Simplify the Second Term
The second term of the expression is x+8/x-4. To simplify this term, we can start by combining the numerator and the denominator.
$\frac{x+8}{x-4} = \frac{x-4+12}{x-4}$
Now, we can see that the x-4 in the numerator can be cancelled out with the x-4 in the denominator, leaving us with:
$1 + \frac{12}{x-4}$
Step 3: Combine the Terms
Now that we have simplified both terms, we can combine them to get the final expression.
$3\left(\frac{1}{1-\frac{4}{x}}\right) - \left(1 + \frac{12}{x-4}\right)$
Step 4: Simplify Further
We can further simplify the expression by combining the fractions.
$\frac{3}{1-\frac{4}{x}} - \frac{x-4+12}{x-4}$
$\frac{3}{1-\frac{4}{x}} - 1 - \frac{12}{x-4}$
$\frac{3}{\frac{x-4}{x}} - 1 - \frac{12}{x-4}$
$\frac{3x}{x-4} - 1 - \frac{12}{x-4}$
Final Expression
The final simplified expression is:
$\frac{3x}{x-4} - 1 - \frac{12}{x-4}$
In conclusion, we have successfully rationalized the expression 3x/x-4 - x+8/x-4 and simplified it to its final form.
