Find the Domain: When the Denominator 2x-6 Affects the Rational Equation

Rational Function Domains with Linear Denominators

What is the domain of the exercise?

5x+82x6=30 \frac{5x+8}{2x-6}=30

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Find the assignment domain
00:03 Assignment domain exists, to ensure we don't divide by 0
00:06 Isolate X to find the assignment domain
00:10 This is the assignment domain, and this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

What is the domain of the exercise?

5x+82x6=30 \frac{5x+8}{2x-6}=30

2

Step-by-step solution

To find the domain of the expression 5x+82x6=30\frac{5x+8}{2x-6} = 30, we need to identify values of xx that make the denominator of the fraction zero.

Step 1: Identify the denominator of the fraction, which is 2x62x - 6.

Step 2: Set the denominator equal to zero to find the values to exclude:

  • Solve the equation 2x6=02x - 6 = 0.
  • Add 6 to both sides: 2x=62x = 6.
  • Divide both sides by 2: x=3x = 3.

Therefore, x=3x = 3 is the value that makes the denominator zero, so it must be excluded from the domain.

Given the choices, the correct answer is x3x \neq 3.

Therefore, the domain of the expression is all real numbers except x=3x = 3.

This implies that the correct choice is:

x3 x \neq 3

3

Final Answer

x≠3

Key Points to Remember

Essential concepts to master this topic
  • Domain Rule: Exclude values that make any denominator equal zero
  • Technique: Set 2x6=0 2x - 6 = 0 and solve: x=3 x = 3
  • Check: Substitute x=3 x = 3 : denominator becomes 2(3)6=0 2(3) - 6 = 0

Common Mistakes

Avoid these frequent errors
  • Solving the entire equation instead of just finding domain restrictions
    Don't solve 5x+82x6=30 \frac{5x+8}{2x-6} = 30 for x = the equation's solution! This gives you where the equation is satisfied, not domain restrictions. Always set only the denominator equal to zero to find excluded values.

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 do we only look at the denominator to find the domain?

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The domain is all possible input values. Since division by zero is undefined, any x-value that makes the denominator zero must be excluded from the domain, regardless of what the numerator equals.

What's the difference between the domain and solving the equation?

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Domain asks: "What x-values are allowed?" Solving asks: "What x-values make the equation true?" These are completely different questions with different answers!

Do I need to worry about the numerator being zero?

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No! A numerator can equal zero - that just makes the whole fraction equal zero. Only when the denominator equals zero do we have an undefined expression.

How do I write the domain properly?

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You can write it as x3 x ≠ 3 or "all real numbers except 3" or in interval notation: (,3)(3,) (-∞, 3) ∪ (3, ∞) .

What if there are multiple fractions in the equation?

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Find the values that make any denominator zero. If you have 1x2+1x+1=5 \frac{1}{x-2} + \frac{1}{x+1} = 5 , then exclude both x = 2 and x = -1 from the domain.

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