# Thread: Solution sets and simplest forms

1. ## Solution sets and simplest forms

i was wondering if anyone could help me out with these two problems?!

expressed in simplest form: 5x+3/x - x-1/2x is??

and the solution set of? 3x+2/x=4x-2/3x+1/3

Thanks for any help!

2. 1) expressed in simplest form: 5x+3/x - x-1/2x is??

So first I'm interpreting this as 5x + 3/x - x - 1/(2x). that is the problem I am solving.

Step 1: bring to a common denominator

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

simplified: 10(x^2)/(2x) + 6/(2x) - 2(x^2)/(2x) - 1/(2x)

(10(x^2) + 6 - 2(x^2) - 1)/(2/x)

Simplify

(8(x^2) + 5)/(2x)

2) and the solution set of? 3x+2/x=4x-2/3x+1/3

Interpretation: 3x + 2/x = 4x - (2/3)x + 1/3 is it (2/3)x or 2/(3x)?

Step 1: Multiply through by 3x to get rid of denominators

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

simplify

9(x^2) + 6 = 12(x^2) - 2(x^2) + x

simplify more

9(x^2) + 6 = 10(x^2) + x

Step 2: subtract left side to get all values on right side

0 = x^2 + x - 6

Step 3: Factor

0 = (x + 3)(x - 2)

so x = -3,2

if it's 2/(3x) then it's the same procedure but the quadratic won't factor nicely, so you use the quadratic formula

hopefully that wasn't too confusing