Solve for x: x(x-5x+6/x)=0

[griffin.811]

5th ed. Algebra book, p.31.

Looking at this question, do I need to distribute the x to both terms in the parentheses first?

I looked at the (5x+6)/x portion and tried to eliminate the x in the denominator, leaving me with 4x+6. Is this illegal, and if so, why?

Finally, is it always the case that we should distribute first? I was trying to follow the PEMDAS rule, which led me to try the above operation.

Thanks

[RonPurewal]

Per the forum rules, please reproduce the entire question as originally written. If there are answer choices, please provide those, too.

We don't always (or even usually) have the books on hand while moderating the forum, so this is not just a random rule; it's essential.
Thanks.

[braidengoodchild]

His subject contains the question verbatim. Guide 2, Chapter 2, Question 2.

[RonPurewal]

Ah, ok. That's not how it's supposed to work, but, fine for this time.

Looking at this question, do I need to distribute the x to both terms in the parentheses first?

As always, ask yourself whether there's a REASON to do the steps you're considering.

Here, if you multiply the x into the parentheses, you'll kill the fraction with denominator x. That's a good reason if I've ever seen one.

[RonPurewal]

I looked at the (5x+6)/x portion and tried to eliminate the x in the denominator, leaving me with 4x+6. Is this illegal, and if so, why?

First, watch your transcription. The heading says x times (x – 5x + 6/x). That looks like three terms.

If it's x(x – (5x + 6)/x), then the extra parentheses are necessary.

If that's what the expression says, then you're making two mistakes:
* You're subtracting instead of dividing. Your 4x is 5x minus x. If you divide 5x by x, you just get 5.
* Yes, you need to divide both top terms by the denominator.

If you just multiply the x into the parentheses, this denominator will disappear, and then this will become a non-issue.

[RonPurewal]

Finally, is it always the case that we should distribute first? I was trying to follow the PEMDAS rule, which led me to try the above operation.

Thanks

PEMDAS is solely for operations written from left to right with no parentheses.
It doesn't apply to fraction bars. Using a fraction bar is exactly the same as putting parentheses around the top and parentheses around the bottom.

Ironically, PEMDAS is the reason why your heading actually means
x • ((x) – (5x) + (6/x)), rather than what you want it to mean.