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

Question

Determine the area of the domain without solving the expression:

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

00:00 Find the domain of substitution

00:02 Substitution domain exists, to ensure we don't divide by 0

00:05 Let's isolate X to find the domain of substitution

00:08 This is one substitution domain, now let's find the second one

00:11 This is the second substitution domain, the domain of substitution is both of them together

00:14 And this is the solution to the 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 $x \neq 2, x \neq 6$ .

$x≠2,x≠6$