Solving Rational Equations: What is the Missing Step?

[Mathnomalous]

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?

 

[physicsman2]

Why would you think you're missing something?

 

[Mathnomalous]

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.

 

[hgfalling]

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.

 

[Mathnomalous]

Well, that confirms my suspicion. This is the first exercise in this assignment that has no solutions.

Thank you for your time and assistance!

 

[physicsman2]

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.