SOLUTION: Please help me solve this equation: {{{ x/(x+1)-2=3/(x-3) }}}

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Question 268825: Please help me solve this equation: +x%2F%28x%2B1%29-2=3%2F%28x-3%29+
Found 2 solutions by mananth, jsmallt9:
Answer by mananth(16946) About Me  (Show Source):
You can put this solution on YOUR website!
x/(x+1)-2=3/(x-3)
x-2x-2 = 3(x+1)/(x-3)
-x-2= 3x+3 /x-3
(x-3(-x-2)= 3x+3
-x^2-2x+3x+6=3x+3
-x^2-2x+3=0
x^2-2x-3=0
x^2-3x+x-3=0
x(x-3)+1(x-3)=0
(x-3)(x+1)=0

Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
(NOTE: There is an error in a solution provided by another tutor.)

+x%2F%28x%2B1%29-2=3%2F%28x-3%29+

We'll start by eliminating the fractions. This is done by multiplying both sides of the equation by the lowest common denominator (LCD). The LCD here is (x+1)(x-3). Multiplying both sides by the LCD we get:
%28x%2B1%29%28x-3%29%28+x%2F%28x%2B1%29-2%29=%28x%2B1%29%28x-3%29%283%2F%28x-3%29%29+
On the left side we need to use the Distributive Property:

Now we can cancel factors:

%28x-3%29%28+x%29-%28x%2B1%29%28x-3%292=%28x%2B1%29%283%29
Now we can multiply. (Pay close attention to the minus sign in the middle of the left side!)
x%5E2+-+3x+-+%28x%5E2+-2x+-+3%292+=+3x%2B3
x%5E2+-+3x+-+%282x%5E2+-4x+-+6%29+=+3x%2B3
x%5E2+-+3x+%2B+%28-2x%5E2%29+%2B4x+%2B+6+=+3x%2B3
-x%5E2+%2Bx+%2B+6+=+3x+%2B+3
The fractions are gone. Now we have a quadratic equation to solve. We start by making one side zero (by subtracting 3x and 3 from each side):
-x%5E2+-2x+%2B+3+=+0
To make the factoring easier I'm going to multiply both sides by -1:
x%5E2+%2B2x+-+3+=+0
Now it factors into
%28x%2B3%29%28x-1%29+=+0
According to the Zero Product Property this product can be zero only if one of the factors is zero. So:
x%2B3+=+0 or x-1+=+0
Solving these we get:
x+=+-3 or x+=+1

We should check our answers. It is more than a good idea to check our answers in this problem because when we multiplied both sides by the LCD and, depending on the value for x, the LCD might be zero. And whenever you multiply both sides by something that might be zero you may get answers that don't actually fit the original equation. So we must check.
+x%2F%28x%2B1%29-2=3%2F%28x-3%29+
Checking x = -3:
+%28-3%29%2F%28%28-3%29%2B1%29-2=3%2F%28%28-3%29-3%29+
+%28-3%29%2F%28-2%29-2=3%2F%28-6%29+
+3%2F2-+4%2F2=-1%2F2+
+1%2F2=-1%2F2+ Check!

Checking x = 1:
+%281%29%2F%28%281%29%2B1%29-2=3%2F%28%281%29-3%29+
+1%2F2-2=3%2F%28-2%29+
+1%2F2-+4%2F2=-3%2F2+
+-3%2F2=-3%2F2+ Check!