How do you solve #((x1) / (x^2x12))  4/(x^26x+8) = x/(x^2+x6)#?
Start by factoring the denominators to see what our denominator will have to be to put on equivalent bases.
Applying a common denominator through multiplication:
Now that the denominators have been removed, we can solve as a regular equation.
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To solve the equation ((x1) / (x^2x12))  4/(x^26x+8) = x/(x^2+x6), we can follow these steps:

Factorize the denominators of each fraction:
 (x^2x12) can be factored as (x4)(x+3)
 (x^26x+8) can be factored as (x2)(x4)
 (x^2+x6) can be factored as (x2)(x+3)

Rewrite the equation with the factored denominators:
 ((x1) / ((x4)(x+3)))  (4 / ((x2)(x4))) = (x / ((x2)(x+3)))

Find a common denominator for all the fractions, which is (x2)(x4)(x+3).

Multiply each term by the common denominator to eliminate the fractions:
 (x1)(x2)(x4)  4(x2)(x+3) = x(x4)(x+3)

Expand and simplify the equation:
 (x^3  7x^2 + 14x  8)  (4x^2  2x  24) = x^3  4x^2 + 3x

Combine like terms on both sides of the equation:
 x^3  7x^2 + 14x  8  4x^2 + 2x + 24 = x^3  4x^2 + 3x

Simplify the equation further:
 x^3  11x^2 + 16x + 16 = x^3  4x^2 + 3x

Subtract x^3 from both sides of the equation to eliminate the x^3 terms:
 11x^2 + 16x + 16 = 4x^2 + 3x

Combine like terms on both sides of the equation:
 11x^2 + 4x^2 + 16x  3x + 16 = 0

Simplify the equation further:
 7x^2 + 13x + 16 = 0
 At this point, the equation cannot be factored easily, so we can use the quadratic formula to find the solutions for x:
 x = (b ± √(b^2  4ac)) / (2a)
Plugging in the values from the quadratic equation, we get:
 x = (13 ± √(13^2  4(7)(16))) / (2(7))
Simplifying further:
 x = (13 ± √(169 + 448)) / (14)
 x = (13 ± √617) / (14)
Therefore, the solutions for x are:
 x = (13 + √617) / (14)
 x = (13  √617) / (14)
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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