SOLUTION: I really need your help, Solve for the following equation x..... 7/x-4+1/x+4=-7x/x^2-16

Question 426432: I really need your help, Solve for the following equation x.....
7/x-4+1/x+4=-7x/x^2-16
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!

I think most would agree that an equation without fractions is easier than an equation with fractions. So we are going to start by eliminating the fractions.

Fractions can be eliminated from an equation by

Finding the Lowest Common Denominator (LCD) of all the fractions on both sides of the equation.
Multiply both sides of the equation by the LCD.

To find a LCD we need to now the factors of the denominators. The first two denominators will not factor. But the third one is a difference of squares and will factor:

Now that we can see the denominators in factored form we can figure out the LCD. The LCD will be the smallest product that includes all the factors of all the denominators. In this case the LCD is (x+4)(x-4). Multiplying both sides of the equation by the LCD:

On the left side, because it is two terms, we must use the Distributive Property:

All the denominators cancel!

leaving

Simplifying we get:
7x + 28 + x - 4 = -7x
8x + 24 = -7x
Now we can solve for x. Subtracting 8x from each side we get:
24 = -15x
Dividing by -15 we get:

Reducing we get:

The only thing left to do is to check your answer. Whenever you multiply both sides by an equation by an expression that could be zero, like we did when we multiplied by the LCD, you must check your answers. The main thing to check is that your answers do not make a denominator zero. I think we can easily see that if x = -8/5 that the denominators, x+4, x-4 and (x+4)(x-4), will not be zero. So our answer looks good. (If a denominator had been zero when x = -8/5, then we would have to reject the solution.) The full check will tell us if we made a mistake. You are welcome to finish the check if you like.

P.S. Please put multiply term numerators and denominators in parentheses:
7/(x-4)+1/(x+4)=(-7x)/(x^2-16)
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