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Equations are easier if there are no fractions, I think you'll agree. So we'll start by eliminating the fractions. The asiest way to eliminate the fractions in an equation is to multiply both sides by the Lowest Common Denominator (LCD). And to find the LCD we need to factor the denominators:
The LCD is the product of all the different factors. So the LCD here is:
(x+2)x
This is what we will multiply both sides by:
On the left side we need to use the Distributive Property:
Now we can cancel:
leaving:
which simplifies as follows:
Without the fractions, this is a very simple equation to solve. It is a quadratic equation so we want one side equal to zero. So subtract 12 from each side:
Now we factor:
3x(x + 2) = 0
From the Zero Product Property we know that this product can be zero only if one of the factors is zero. So:
3x = 0 or x+2 = 0
Solving these we get:
x = 0 or x = -2
With equations where the variable is in one or more denominators, it is important to check your answers. We must make sure no denominators are zero! Always check with the original equation.
Checking x = 0:
which simplifies to:
As you can see, two of the denominators are zero. For this reason we must reject x = 0 as a solution. (If even only one denominator was zero we would still reject the solution.)
Checking x = -2:
which simplifies to:
Again we get zero denominators! We must reject this solution, too.
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