SOLUTION: Solve r+(r^2-5)/(r^2-1)=(r^2+r+2)/(r+1) Please I need help with this! Thanks!

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Question 825741: Solve
r+(r^2-5)/(r^2-1)=(r^2+r+2)/(r+1)
Please I need help with this! Thanks!
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
r%2B%28r%5E2-5%29%2F%28r%5E2-1%29=%28r%5E2%2Br%2B2%29%2F%28r%2B1%29
Solving equations with fractions is harder than solving equations without fractions. So we will make the problem easier if we eliminate the fractions as soon as possible.

Fractions in an equation can be eliminated by...
  1. Find the lowest common denominator (LCD) of all the fractions (on both sides of the equation).
  2. Multiply both sides of the equation by the LCD.
The first denominator will factor:
r%5E2-1+=+%28r%2B1%29%28r-1%29
The second denominator, r+1, does not factor. Looking at the second denominator and the factored first denominator, we should be able to figure out that the LCD is: (r+1)(r-1). So we will multiply both sides by (r+1)(r-1):

First we must use the Distributive Property on the left side:

Now as we multiply, each denominator will cancel with some part of (r+1)(r-1):

Leaving:
%28r%2B1%29%28r-1%29%28r%29%2B+1%2A%28r%5E2-5%29=%28r-1%29%28r%5E2%2Br%2B2%29
Now we solve. First we simplify:

r%5E3-r%2B+r%5E2-5=r%5E3%2Br%5E2%2B2r%2B%28-r%5E2%29%2B%28-r%29%2B%28-2%29
r%5E3%2B+r%5E2-r-5=r%5E3%2Br%2B%28-2%29
Next we get a zero on one side. Subtracting the entire right side from both sides:
r%5E2-2r-3=0
Now we factor:
(r-3)(r+1) = 0
From the Zero Product Property:
r-3 = 0 or r+1 = 0
Solving these we get:
r = 3 or r = -1

Last we check. This is not optional! When both sides of an equation are multiplied by something that might be zero, like (r+1)(r-1), then a check is required. One must make sure that the solution does not make a factor of (r+1)(r-1) equal to zero.

A quick visual check should tell us that if r = 3 (one solution) then neither (r+1) nor (r-1) will be a zero. So this solution checks!

A quick visual check should tell us that if r = -1 (the other solution) then (r+1) will be a zero! So this solution fails the check and must be rejected!

So there is only one solution to this problem: r = 3

P.S. This equation has rational expressions. It is not a rational function. It should be posted in the "polynomials, rational expressions ..." category.