How do you solve x /( 2x + 1 ) + ( 1 / 4 ) = 2 / (2x + 1) and find any extraneous solutions?

2 Answers
Nov 7, 2017

Solution: 1 1/6 and no extraneous root.

Explanation:

x/(2x+1)+1/4= 2/(2x+1) Multiplying by 4(2x+1) on

both sides we get , 4x+(2x+1)= 8 or 6x= 7or x=7/6

Check: L.H.S =x/(2x+1)+1/4= (7/6)/(2*7/6+1) +1/4

= (7/cancel6)/(20/cancel6)+1/4

=7/20+1/4 = 12/20= 3/5

R.H.S =2/(2x+1) = 2/(2*7/6+1)=2/(20/6)=12/20=3/5

L.H.S=R.H.S , no extraneous root.

Solution: x= 7/6= 1 1/6 [Ans]

Nov 7, 2017

x=7/6

Explanation:

"since fractions on both sides of the equation have a"
color(blue)"common denominator "" we can combine them"

"subtract "2/(2x+1)" from both sides"

x/(2x+1)-2/(2x+1)+1/4=0

"subtract "1/4" from both sides"

x/(2x+1)-2/(2x+1)cancel(+1/4)cancel(-1/4)=0-1/4

rArr(x-2)/(2x+1)=-1/4larrcolor(blue)"combining fractions"

"using the method of "color(blue)"cross-multiplying"

•color(white)(x)a/b=c/drArraxxd=bxxc

"attaching the negative sign to 1 "

rArr-1(2x+1)=4(x-2)

rArr-2x-1=4x-8larrcolor(blue)"distributing"

"subtract 4x from both sides"

-2x-4x-1=cancel(4x)cancel(-4x)-8

rArr-6x-1=-8

"add 1 to both sides"

-6xcancel(-1)cancel(+1)=-8+1

rArr-6x=-7

"divide both sides by "-6

(cancel(-6) x)/cancel(-6)=(-7)/(-6)

rArrx=7/6" is the solution"

"there are no extraneous solutions"