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

\( 2x+\frac{6}{x}=18 \)

What is the domain of the above equation?

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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