How Do You Solve Complex Fraction Equations?

[The_big_dill]

Homework Statement

Solve for X.

Homework Equations

(3 / x+2) - (1 / x) = 1 / 5x

The Attempt at a Solution

(NOTE: I always had difficulties with fractions)

(3x - x - 2 / x² + 2) - (1 / 5x) = 0

(2x - 2 - x² - 2) / 5x³ + 10x

(-x² + 2x - 4) / 5x³ + 10x

^ I attempted a few variations of the above. I have also attempted factoring and quadratic equation, but none of the values work when i check by substituting them into the equation.

What am i doing wrong? (Where?)

 

[JonF]

What is the least common multiple of: x+2, x, 5x?

 

[The_big_dill]

What is the least common multiple of: x+2, x, 5x?

I hope you mean product of x + 2, x. And product of 5x. (Just checking my understanding, PLEASE correct me if i am getting something wrong, this should be math review for me).

In any case,

x = -2

x = 0

x = 1/5On a side note... Was i making this question more difficult then it really should be? Do i just look at wherever X is (despite all other values) and find out all zeros?

 

[JonF]

The LCM for polynomials works the same way as it does for numbers.

If I asked you for the LCM for 4 6 and 15 it would be 60, since it’s the smallest number that 4,6, and 15 all are factors of.

So what is the smallest polynomial that has: x+2, x, 5x all as factors? (Notice that these polynomials are our denominators)

Once we have our LCM we can make all of our fractions have like denominators and we will be much closer to solving this problem!

 

[The_big_dill]

5x³ + 10x²

 

[JonF]

Perfect! Let’s leave it as 5x(x+2) for a second to make our life easy.

So if I wanted to make each of these fractions denominators 5x(x+2) how would I do it?
3/(x+2)
1/x
1/5x

If you’re not sure, ask yourself how you would do it with numbers. How would I make these fractions denominator 60?
3/4
1/6
1/15

I’ll do the first one to help you out. I know 4*15 = 60, so I’ll multiply the top and bottom by 15. So 3/4 = (3*15)/(4*15) = 45/60

 

[The_big_dill]

Perfect! Let’s leave it as 5x(x+2) for a second to make our life easy.

So if I wanted to make each of these fractions denominators 5x(x+2) how would I do it?
3/(x+2)
1/x
1/5x

If you’re not sure, ask yourself how you would do it with numbers. How would I make these fractions denominator 60?
3/4
1/6
1/15

I’ll do the first one to help you out. I know 4*15 = 60, so I’ll multiply the top and bottom by 15. So 3/4 = (3*15)/(4*15) = 45/60

What i do to denominator i must do to numerator.

(3 * 5x) - (1 * (x + 2)) / 5x (x + 2)

 

[JonF]

Exactly (well as far as multiplication is concerned)! so if i wanted 3/(x+2) to have a denominator of 5x(x+2) what would I need to do?

 

[The_big_dill]

Exactly (well as far as multiplication is concerned)! so if i wanted 3/(x+2) to have a denominator of 5x(x+2) what would I need to do?

Not very clear, specifically because (from my understanding) i would give you the same response as my last post.

 

[JonF]

You multiplied the denominator by 5x. It looks like you did a lot of other stuff to the top. If we multiply the denominator by 5x all we do the top is multiply it by 5x.

So 3/(x+2) = (3*5x)/[5x*(x+2)]

Go ahead and try the other 2