anonymous 3 years ago Simplify

1. anonymous

$\frac{ 2x }{ 4\pi }+ \frac{ 1-x }{ 2 }$

2. hartnn

is that really $$\pi$$ in the denominator of 1st fraction ?

3. anonymous

yes

4. hartnn

now we have to make the denominator common , so we need to multiply and divide by 2$$\pi$$ in 2nd fraction to get 4$$\pi$$as common denominator . $$\huge\frac{2x}{4\pi}+\frac{2\pi(1-x)}{4\pi}$$ now can u solve further ??

5. anonymous

i got up to $\frac{ x-pix }{ 2\pi} = 1/2$

6. anonymous

i mean -1/2

7. hartnn

nopes, $$\huge\frac{2x}{4\pi}+\frac{2\pi(1-x)}{4\pi}=\frac{2x+2\pi(1-x)}{4\pi}=\frac{x+\pi-\pi x}{2\pi}$$ u can't simplify it further.

8. anonymous

i forgot one important thing. the equation is equal to zero. and i'm solving for x

9. hartnn

okay, so it will be $$\huge x+\pi-\pi x=0 \implies x(1-\pi)=-\pi \\\huge x=\frac{-\pi}{1-\pi}or\frac{\pi}{\pi-1}$$

10. anonymous

so was my step right so far?

11. hartnn

u got -1/2, but its actually, pi/(pi-1)

12. anonymous

equal to zero? my teacher did the same thing as me.. but i didn't write the last part leading towards the answer.

13. anonymous

i crossed multiply $\frac{ 4x }{ 8\pi } + \frac{ 4\pi - 4pix }{ 8\pi } = 0$

14. anonymous

i split the equation up also

15. hartnn

the first term has 'x'

16. anonymous

the i got $\frac{ x }{ 2\pi } - \frac{ \pi }{ 2\pi } = -1/2$

17. anonymous

the $\frac{ \pi }{ 2 reduces to 1/2 and i brought it over... 18. hartnn but u cannot cancel pi, the first term had x in it, not pi. 19. anonymous yes... and i'm solving for it. i haven't found the answer further than that step 20. anonymous what are you talking about cancel? i didn't cancel anything other than the pis... which was pi/2pi... they're in common? this part i know for sure is right. i don't know how to get after that 21. hartnn how did u get -1/2 ...and where did your 8 go from denominator ? 22. anonymous that was a typo. and i reduced \[\frac{ \pi }{ 2\pi }$ to get 1/2 and then i subtracted it on both sides so the equation is equal to -1/2

23. anonymous

i'm basically trying to isolate the x as much as possible

24. hartnn

$$\huge \frac{ 4x }{ 8\pi } + \frac{ 4\pi - 4\pi x }{ 8\pi } = 0$$ u had this correct. then u separated denominator u should get, $$\large \frac{x}{2\pi}+\frac{1}{2}-\frac{x}{2}=0$$ but this is not the best way to isolate x.

25. anonymous

yea, but it's easier for me to visualize it. instead of clutter i wanted to separate the fractions

26. hartnn

ok, so continuing with separating the fraction, you can cancel all the 2's in the denominator(equivalent to multiplying 2 on both sides.) then shifting +1to other side, u get $$\large x(\frac{1}{\pi}-1)=-1$$ which again gives, $$\large x=\frac{-1}{1/\pi-1}=\frac{\pi}{\pi-1}$$

27. hartnn

ok, so continuing with separating the fraction, you can cancel all the 2's in the denominator(equivalent to multiplying 2 on both sides.) then shifting +1to other side, u get $$\large x(\frac{1}{\pi}-1)=-1$$ which again gives, $$\large x=\frac{-1}{1/\pi-1}=\frac{\pi}{\pi-1}$$

28. anonymous

so i multiply the top and bottom by 2pi to get rid of it in the denominator?

29. anonymous

where do you get +1?

30. hartnn

yes, u can do that also, that will lead to same answer.

31. hartnn

as i said, i multiplied both sides by 2

32. hartnn

as i said, i multiplied both sides by 2

33. anonymous

$\frac{ 2x }{ 4\pi }+ \frac{ 1-x }{ 2 }$ is not an equation. there is nothing to "solve" for you can add however, by finding the lcd is $$4\pi$$ and adding

34. anonymous

you would get $\frac{2x+2(1-x)}{4\pi}$

35. anonymous

i got x = $\frac{ \pi }{ 1-\pi }$

36. anonymous

- pi / 1-pi = pi / 1+pi

37. hartnn

there must be slight error in minus sign, its pi/(pi-1)

38. anonymous

no, i subtracted 1/2 from both sides and it was -1/2

39. anonymous

then i multiplied both sides by 2pi which made it -pi.

40. anonymous

so i got $x - pix = -pi$

41. hartnn

both those steps are correct. u get x(1-pi)=-pi x=-pi/(1-pi) = pi/(pi-1)

42. anonymous

then i factored out the x. $x(1-\pi) = -\pi$ $x = \frac{ -\pi }{ 1-\pi }$

43. hartnn

yes so its -pi/(1-pi) which is same as pi/(pi-1)

44. anonymous

are you sure? if you take out the negative, wouldn't the denominator be 1+ pi?

45. hartnn

nopes it would not be 1+pi. it will be pi-1 -(1-pi) = pi-1 yes, sure.

46. anonymous

ok i see you flipped it around. lol