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

Q: Why don't we exclude x=0 even though it's in the numerator?

When x=0, the numerator 7x becomes zero, making the fraction equal zero (not undefined). Only denominators equal to zero create undefined expressions that must be excluded from the domain.

Q: Do I need to solve the equation to find the domain?

No! The domain depends only on where the expression is undefined, not on the solutions. Just find where each denominator equals zero and exclude those values.

Q: Can the domain change if I simplify the fractions first?

Be careful! If you cancel common factors, you might accidentally remove domain restrictions. It's safest to find the domain from the original expression before simplifying.

Q: How do I write domain restrictions properly?

Use the format $ x ≠ a, x ≠ b $ to list all excluded values, or write as intervals: $ (-∞,2) ∪ (2,6) ∪ (6,∞) $.

Domain Analysis with Rational Functions

Determine the area of the domain without solving the expression:

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

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

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.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Determine the area of the domain without solving the expression:

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

Step-by-step solution

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 $x \neq 2, x \neq 6$ .

Final Answer

$x≠2,x≠6$

Key Points to Remember

Essential concepts to master this topic

Domain Rule: Exclude all values making denominators equal zero
Technique: Set each denominator to zero: x-2=0 gives x=2
Check: Verify exclusions don't make original fractions undefined ✓

Common Mistakes

Avoid these frequent errors

Including numerator zeros as domain restrictions
Don't exclude x=0 just because 7x has x in numerator = wrong domain! Numerators equal zero create zeros, not undefined values. Always exclude only denominator zeros: x=2 and x=6.

Practice Quiz

Test your knowledge with interactive questions

Select the the domain of the following fraction:

$\frac{6}{x}$

FAQ

Everything you need to know about this question

Why don't we exclude x=0 even though it's in the numerator?

When x=0, the numerator 7x becomes zero, making the fraction equal zero (not undefined). Only denominators equal to zero create undefined expressions that must be excluded from the domain.

Do I need to solve the equation to find the domain?

No! The domain depends only on where the expression is undefined, not on the solutions. Just find where each denominator equals zero and exclude those values.

What if the equation had more fractions?

The process stays the same! Set each denominator equal to zero, solve for x, and exclude all those values from the domain. Each fraction contributes its own restrictions.

Can the domain change if I simplify the fractions first?

Be careful! If you cancel common factors, you might accidentally remove domain restrictions. It's safest to find the domain from the original expression before simplifying.

How do I write domain restrictions properly?

Use the format $x ≠ a, x ≠ b$ to list all excluded values, or write as intervals: $(-∞,2) ∪ (2,6) ∪ (6,∞)$ .

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Algebraic Expressions questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime

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

Domain Analysis with Rational Functions

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why don't we exclude x=0 even though it's in the numerator?

Do I need to solve the equation to find the domain?

What if the equation had more fractions?

Can the domain change if I simplify the fractions first?

How do I write domain restrictions properly?

🌟 Unlock Your Math Potential

More Questions

Domain of a Function

Determine the Domain of the Function: √(5x²+2) over 10

Determine the Domain of: √(3x²+7)/12

Analyzing Domain in Rational Equation: 14/x - 6x = 2/(x-5)

Explore Rational Equation Domains: Find the Area of (x/(5x-69) = 2/(x-1)

Domain

Solve for Domain Area: 9(4x-5/x) = 20(3x-6/(x+1)) Rational Equation

Find the Domain of 7/(x+5) = 6/(13x): Rational Equation Analysis