Question

How do you solve `\frac { 2x } { 2} + \frac { 2} { x } = \frac { 6} { 2x }`?

Answer 1

`x={-1,1}`

Explanation:

`"given "(2x)/2+2/x=6/(2x)`

`color(red)(x/x)*(2x)/2+color(blue)(2/2).2/x=6/(2x)`

`(2x^2)/(2x)+4/(2x)=6/(2x)`

`1/(cancel(2x))(2x^2+4)=6/(cancel(2x))`

`2x^2+4=6`

`2x^2=6-4`

`cancel(2)x^2=cancel(2)`

`x^2=1`

`sqrt x^2=sqrt 1`

`x={-1,1}`

Answer 2

`x = +-1`

Explanation:

In an equation which has fractions, you can always get rid of the fractions immediately by multiplying each term by the LCM of the denominators so you can cancel the denominators.

In this case it will be `color(blue)(2x)`.

`(2x)/2 +2/x = 6/(2x)`

`(color(blue)(cancel2x)xx2x)/cancel2 +(color(blue)(2cancelx)xx2)/cancelx = (color(blue)cancel(2x)xx6)/cancel(2x)`

`2x^2 +4 = 6`

`2x^2 = 6-4`

`2x^2 =2" "larr div 2`

`x^2 = 1" "larr` find the square root of both sides

`x = +-1`

Note that bother of these can be used as solutions because they will not make the denominators equal to 0.