Exploring Domains in Rational Fractions: (4/(x-2))×(7x/(x-6))=2

Question

Determine the area of the domain without solving the expression:

$(\frac{4}{x-2})\times(\frac{7x}{x-6})=2$

00:15 Let's find the domain for substitution.

00:18 Remember, the domain is where we make sure we're not dividing by zero.

00:24 First, we'll isolate X to figure out the domain for substitution.

00:29 Great! That's the first domain. Let's find the second one now.

00:34 Here's the second substitution domain. Combine both, and you've got the full domain.

00:39 And there you have it. That's how we solve this question.

To solve this problem, we'll determine where the given expression is undefined:

Step 1: Identify where the first fraction, $\frac{4}{x-2}$ , is undefined. This fraction is undefined when its denominator is zero: $x-2 = 0$ . Thus, $x = 2$ .
Step 2: Identify where the second fraction, $\frac{7x}{x-6}$ , is undefined. This fraction is undefined when its denominator is zero: $x-6 = 0$ . Thus, $x = 6$ .
Step 3: The expression is undefined at $x = 2$ and $x = 6$ .

Therefore, the domain of the expression excludes $x = 2$ and $x = 6$ .

The correct domain restriction is $x \neq 2, x \neq 6$ .

$x≠2,x≠6$