How do you solve #\frac { 2} { 3} = \frac { x + 3} { 4x } + \frac { x - 6} { 6x }#?

1 Answer
Ricardo A.
May 3, 2017

#x=-1#

Explanation:

First we add the two fractions on the right by making them have the same denominator. We do that by finding the smallest multiple between the two numbers and multiplying the top and bottom of the fractions to get there. In this case, we have 4 and 6 and their smallest common multiple is 12. So we need to multiply 1 of the fractions by 3 and the second by 2 like this:

#2/3=(3*(x+3))/(3*4x)+(2*(x-6))/(2*6x)#

#2/3=(3x+9)/(12x)+(2x-12)/(12x)#

That allows us to join the two fractions into one and sum the nominators.

#2/3=(3x+9 +2x-12)/(12x)#

#2/3=(5x-3)/(12x)#

Now we can multiply both sides by 3.

#3*2/3=(5x-3)/(12x)*3#

#cancel(3)*2/cancel(3)=(5x-3)/(""_4 cancel(12)x)*cancel3#

Leaving us with:

#2=(5x-3)/(4x)#

Multiply both sides by 4x

#4x*2=(5x-3)/cancel(4x)*cancel(4x)#

#8x=5x-3#

Subtract 5x from both sides

#8x-5x=-3#

#3x=-3#

divide both sides by 3

#(cancel(3)x)/cancel3=-3/3#

#x=-1#

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