math help please?

Question

solve and check equation for x

x/x-2 - x+1/x = 8/x^2-2x

Answer 1

x/ x-2 - x+1/ x=8/ x^2-2x
x/x-2 -x+1/x =8/x(x-2)

x^2-(x+1)(x-2)=8
x^2-x^2+x+2=8
x+2=8
x=6

Answer 2

I assume you mean:
.. x ..... x + 1 ........ 8
------ - --------- = ---------
x - 2 ...... x ........ x² - 2x

On the left, get a common denominator of x(x - 2). Multiply the first fraction by x on top and bottom. Multiply the second equation by (x - 2) on top and bottom:

.... x² ..... (x + 1)(x - 2) ........ 8
--------- - ----------------- = ---------
x(x - 2) ... x(x - 2) ...........x² - 2x

Now write the fractions on the left as one fraction:
x² - (x + 1)(x - 2) ........ 8
----------------------- = ---------
... ... x(x - 2) ............ x² - 2x

If you expand out the denominator, you'll see it is the same on both sides:

x² - (x + 1)(x - 2) ........ 8
----------------------- = ---------
....... x² - 2x ............ x² - 2x

So cancel them:
x² - (x + 1)(x - 2) = 8

Expand everything out:
x² - (x² - 2x + x - 2) = 8
x² - (x² - x - 2) = 8
x² - x² + x + 2 = 8
x + 2 = 8
x = 6

Double-check your answer:
.. 6 ..... 6 + 1 ........ 8
------ - --------- = ---------
6 - 2 ...... 6 ........ 6² - 2(6)

Left side:
6/4 - 7/6
= 3/2 - 7/6
= 9/6 - 7/6
= 2/6
= 1/3

Right side:
8 / (36 - 12)
= 8 / 24
= 1/3

Yes, they balance so x = 6 is the solution.

Answer 3

Multiply them so that they all have the same denominator, then you can simplify from there.

So, ((x)(x)(x^2-2x))/(x-2(x)(x^2-2x)) - ((x)(x-2)(x^2-2x))/((x)(x-2)(x^2-2x)) = ((8)(x-2)(x))/((x^2-2x)(x-2)(x))