Question 5069
{{{x/(x-1)+2/(x-5)=-4/(x-1)(x-5)}}}
First of all we have to make a restriction to avoid division by 0. x can not be 1 or 5. (x-1)=1-1=0, (x-5)=5-5=0
Now you have to find the LCD which is (x-1)(x-5). Now find equivalent fractions with a common denominator. The advantage here is that by using the LCD you clear the equation of all fractions.
{{{x/(x-1)*(x-5)/(x-5)+2/(x-5)*(x-1)/(x-1)=-4/(x-1)(x-5)}}}
{{{x(x-5)+2(x-1)=-4}}} Use the distributive property to get rid of the parentheses.
{{{x^2-5x+2x-2=-4}}} add 4 to both sides and combine like terms.
{{{x^2-3x+2=0}}} Change the middle term into 2 numbers whose sum is -3 and whose product is 2. (-2,-1) -2+(-1)=-3, -2(-1)=2
{{{x^2-2x-x+2=0}}} Factor by grouping
{{{x(x-2)-1(x-2)=0}}} Combine like terms
{{{(x-1)(x-2)=0}}}
Solve for x
a)x-1=0, x=1
b)x-2=0, x=2
Since we made a restriction you can see that solution a) cannot be correct.
So, (2) is the solution set.
Check it:
{{{2/(2-1)+2/(2-5)=-4/(2-1)(2-5)}}}
{{{2/1+2/-3=-4/-3}}}
{{{2/1-2/3=4/3}}} Use the LCD (3)
{{{6-2=4}}}
{{{4=4}}}
Hope it helps