How do you solve #x/(2x-6)=2/(x-4)#?

1 Answer
Apr 14, 2017

Restrict the domain so that the solutions do not cause division by zero in the original equation.
Multiply both sides by both denominators.
Solve the resulting quadratic.
Check.

Explanation:

Given: #x/(2x-6)=2/(x-4)#

Restrict the values of x so that any solutions that would cause division by zero are discarded:

#x/(2x-6)=2/(x-4);x!=3,x!=4#

Multiply both sides of the equation by #(2x-6)(x-4)#

#(2x-6)(x-4)x/(2x-6)=(2x-6)(x-4)2/(x-4);x!=3,x!=4#

Please observe how the factors cancel:

#cancel(2x-6)(x-4)x/cancel(2x-6)=(2x-6)cancel(x-4)2/cancel(x-4);x!=3,x!=4#

Here is the equation with the cancelled factor removed:

#(x-4)x=(2x-6)2;x!=3,x!=4#

Use the distributive property on both sides:

#x^2-4x=4x-12;x!=3,x!=4#

We can put the quadratic into standard form by adding #12-4x# to both sides:

#x^2-8x+12=0;x!=3,x!=4#

This factors and the restrictions can be dropped:

#(x - 2)(x-6)=0#

#x = 2 and x = 6#

Check:

#2/(2(2)-6)=2/(2-4)#
#6/(2(6)-6)=2/(6-4)#

#2/(-2)=2/(-2)#
#6/6=2/2#

#-1=-1#
#1=1#

This checks.