Solving Rational Equations: What is the Missing Step?

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Homework Help Overview

The problem involves solving a rational equation for the variable x, specifically the equation 3/2 + 2/2x-4 = 1/x-2. The original poster expresses confusion about a potential missing step in their solution process.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to solve the equation by finding a common denominator and simplifying, but questions whether they have overlooked something important. Other participants question the validity of the solution x = 2, noting that it leads to undefined denominators.

Discussion Status

The discussion has revealed that the solution x = 2 is not valid due to it making the denominators zero. Participants have explored the implications of this and confirmed that the equation ultimately has no solutions.

Contextual Notes

There is mention of different methods used by the professor in class, which adds to the original poster's confusion regarding the solution process. The discussion highlights the importance of considering the domain of the function when solving rational equations.

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Homework Statement



Solve for x.

Homework Equations



3/2 + 2/2x-4 = 1/x-2

The Attempt at a Solution



LCD = 2(x-2)

3/2 + 2/2x-4 = 1/x-2 Mult. all terms by 2(x-2)

3x - 6 + 2 = 2

3x = 6

x = 2

What am I missing here?
 
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Why would you think you're missing something?
 
No idea, really. I think I followed all the correct steps. I also re-checked my class notes and this happens to be one of the exercises done in class by my professor. However, he solved the equation in a different and longer manner but with the same result (even though there is a (x-2)(x-2) = x^2 - 4 ?).

It's probably the inconsistent solution I obtain that bothers me.
 
Well, x = 2 makes your denominators zero, so that can't be the right answer.

Suppose you reduce the second term of the equation by canceling the 2.

Now you have 3/2 + 1/(x-2) = 1/(x-2).

That equation doesn't make any sense, right? Subtracting 1/(x-2) from both sides gets you 3/2 = 0.

There aren't any solutions to this equation.
 
Well, that confirms my suspicion. This is the first exercise in this assignment that has no solutions.

Thank you for your time and assistance!
 
Yeah, you really weren't missing anything per se, you just didn't make the conclusion that x = 2 isn't in the domain of the function, as hgfalling illustrated.
 

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