Solve 6/(x+5) = 1: Determining the Field of Application

Domain Restrictions with Rational Equations

6x+5=1 \frac{6}{x+5}=1

What is the field of application of the equation?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Find the domain of the function
00:03 According to mathematical laws, division by 0 is not allowed
00:07 Since there is a variable in the denominator, we must ensure it is not equal to 0
00:12 Let's isolate the variable X
00:30 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

6x+5=1 \frac{6}{x+5}=1

What is the field of application of the equation?

2

Step-by-step solution

To solve this problem, we will determine the domain, or field of application, of the equation 6x+5=1 \frac{6}{x+5} = 1 .

Step-by-step solution:

  • Step 1: Identify the denominator. In the given equation, the denominator is x+5 x+5 .
  • Step 2: Determine when the denominator is zero. Solve for x x by setting x+5=0 x+5 = 0 .
  • Step 3: Solve the equation: x+5=0 x+5 = 0 gives x=5 x = -5 .
  • Step 4: Exclude this value from the domain. The domain is all real numbers except x=5 x = -5 .

Therefore, the field of application of the equation is all real numbers except where x=5 x = -5 .

Thus, the domain is x5 x \neq -5 .

3

Final Answer

x5 x\operatorname{\ne}-5

Key Points to Remember

Essential concepts to master this topic
  • Domain Rule: Denominator cannot equal zero in any fraction
  • Method: Set x+5 = 0, solve to get x = -5
  • Check: Verify x ≠ -5 prevents division by zero ✓

Common Mistakes

Avoid these frequent errors
  • Solving the equation before finding domain restrictions
    Don't solve 6/(x+5) = 1 to get x = 1 first and ignore domain = missing critical restrictions! This skips the essential step of identifying forbidden values. Always find domain restrictions by setting denominators equal to zero before solving.

Practice Quiz

Test your knowledge with interactive questions

Solve for X:

\( 6 - x = 10 - 2 \)

FAQ

Everything you need to know about this question

What exactly is the 'field of application' or domain?

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The field of application (or domain) is all the x-values that make the equation mathematically valid. It's the set of all possible inputs that don't break any mathematical rules.

Why can't the denominator be zero?

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Division by zero is undefined in mathematics! When x=5 x = -5 , we get 60 \frac{6}{0} , which has no mathematical meaning.

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

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No! The domain depends only on the structure of the equation, not its solution. Just look at what makes denominators zero.

What if there are multiple fractions in the equation?

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Check every denominator separately! Set each one equal to zero and solve. The domain excludes all values that make any denominator zero.

Is the domain the same as the solution to the equation?

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Not at all! The domain tells you which x-values are allowed, while the solution tells you which x-values actually work. For this problem: domain is x ≠ -5, solution is x = 1.

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