How do you solve #1/x-1/(2x)=2x# ?

I know the solution to
#1/x-1/(2x)=2x#
is
#x=+-0.5#
At least it is the solution in my math book.

But I don't understand how you solve it.
When I tried i got #x=+-0,25# I think.

So I typed the problem into Cymath. And it tells me to do the following:

One. Simplify #1/x-1/(2x)# to #1/(2x)#

#1/(2x)=2x#

Two. Multiply both sides by #2x#

#1=2x*2x#

Three. Simplify #2x*2x# to #4x^2#

#1=4x^2#

Four. Divide both sides by #4#

#1/4=x^2#

Five. Take the square root of both sides.

#+-sqrt(1/4)=x#

So the answer is #x=+-0.5#

But the thing I don't understand is the first step. Why does #1/x-1/(2x)# equals to #1/(2x)#?

It just seems so illogical to me. Is there another way to solve the equation? Or can someone please explain why this is?

2 Answers
May 9, 2018

See the explanation please.

Explanation:

#1/x-1/(2x)=2x=># Left side LCD = #2x# :

#2/(2x)-1/(2x)=2x =># or:

#1/(2x)=2x#

Let's continue:

Multiply both sides by #2x#:

#(2x)/(2x)=2x*2x#

#1=4x^2 =># divide both sides by#4#:

#1/4=x^2 =># take square root of both sides:

#+-1/2=x =># or:

#x=+-0.5#

2nd method:

#1/x-1/(2x)=2x #

To get rid of denominators multiply both sides of equation by the Lowest Common Denominator (LCD) in this case #2x#:

#(2x)/x-(2x)/(2x)=2x*2x =># simplify:

#2-1=4x^2=># simplify:

#1=4x^2 =># continue same as above

May 9, 2018

#x=+-1/2#

Explanation:

Given: #1/x-1/(2x)=2x#

Considering the left hand side only for a moment #1/x-1/(2x)#
We need to make the bottom values (denominators) the same. Multiply by 1 and you do not change the value. However, 1 comes in many forms.

#color(green)([1/xcolor(red)(xx1)]-1/(2x)=2x color(white)("dddd")->color(white)("dddd")[1/xcolor(red)(xx2/2)]-1/(2x)=2x )#

#color(green)(color(white)("ddddddddddddddddddddd")->color(white)("ddddd")ubrace([2/(2x)]color(white)("dd")-1/(2x))=2x)#
#color(green)(color(white)("ddddddddddddddddddddddddddd-dddddd")darr)#

#color(green)(color(white)("ddddddddddddddddddddd")->color(white)("dddddddddd") 1/(2x)color(white)("dddd")=2x)#

Multiply both sides by #color(red)(x)#

#color(green)(color(white)("ddddddddddddddddddddd")->color(white)("ddddd") 1/(2cancel(x))color(red)(xx cancel(x))color(white)("dddd")=2xcolor(red)(xx x))#

#color(green)(color(white)("ddddddddddddddddddddd")->color(white)("ddddddddd")1/2=2x^2#

Divide both sides by 2

#color(green)(color(white)("ddddddddddddddddddddd")->color(white)("ddddddddd")1/4=x^2#

Square root both sides
#color(green)( color(white)("ddddddddddddddddddddd")->color(white)("ddddddd")+-1/2=x#
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
#color(blue)("Check")#

Set #x=+1/2#
#1/x-1/(2x)=2x#

#1/(1/2)-1/(2xx1/2)=2xx1/2# ?

#2-1 = 1 color(red)(larr"True")#

Set #x=-1/2#
#1/x-1/(2x)=2x#

#1/(-1/2)-1/(2xx(-1/2))=2xx(-1/2)# ?

#-2+1 =- 1 color(red)(larr"True")#