Determine the Domain for 22(2/x - 1) = 30

Domain Restrictions with Rational Expressions

22(2x−1)=30 22(\frac{2}{x}-1)=30

What is the domain of the equation above?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Find the domain of substitution
00:03 Domain of substitution exists to ensure we don't divide by 0
00:07 Meaning the denominator in the fraction must be different from 0
00:11 We will use this formula in our exercise
00:16 This is the domain of substitution, 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

22(2x−1)=30 22(\frac{2}{x}-1)=30

What is the domain of the equation above?

2

Step-by-step solution

To find the domain of the given function 22(2x−1)=30 22\left(\frac{2}{x} - 1\right) = 30 , follow these steps:

  • Identify critical terms: The term 2x\frac{2}{x} is undefined when x=0 x = 0 because division by zero is undefined.
  • We need to exclude x=0 x = 0 from the domain to ensure the function remains defined.
  • The correct domain for the equation is all real numbers except x=0 x = 0 .

Thus, the domain of the equation is x≠0 x \neq 0 .

Therefore, the solution to the problem is x≠0 x \neq 0 .

3

Final Answer

x≠0

Key Points to Remember

Essential concepts to master this topic
  • Rule: Division by zero is undefined and must be excluded
  • Technique: Find values making denominators zero: 2x \frac{2}{x} undefined when x = 0
  • Check: Test x = 0 in original equation produces division error ✓

Common Mistakes

Avoid these frequent errors
  • Confusing domain restrictions with equation solutions
    Don't solve the equation and think that answer is the domain restriction = wrong approach! The domain restriction comes from undefined expressions, not from solving. Always identify what makes denominators zero before solving.

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 is domain different from solving the equation?

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Domain tells us which x-values we can use in the equation. Solving the equation tells us which x-values make it true. These are completely different concepts!

What if there are multiple fractions in the equation?

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Check each denominator separately! Find all values that make any denominator zero, then exclude all of them from the domain.

Does the domain affect the solution?

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Yes! If your solution happens to be a value excluded from the domain, then the equation has no solution. Always check this after solving.

How do I write domain restrictions properly?

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Use ≠ notation for single restrictions like x≠0 x ≠ 0 . For multiple restrictions, list them all: x≠0,x≠2 x ≠ 0, x ≠ 2 .

What about square roots or other operations?

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Each operation has its own restrictions:

  • Fractions: denominator ≠ 0
  • Square roots: expression under radical ≥ 0
  • Logarithms: argument > 0

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