# What is the least common multiple for #\frac{x}{x-2}+\frac{x}{x+3}=\frac{1}{x^2+x-6}# and how do you solve the equations?

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To find the least common multiple (LCM) for the given equation, we need to first find the common denominator. The denominators in the equation are (x-2), (x+3), and (x^2+x-6).

The LCM of these three denominators is (x-2)(x+3)(x+2).

To solve the equation, we multiply both sides of the equation by the LCM to eliminate the denominators. This gives us:

(x-2)(x+3)(x+2) * \frac{x}{x-2} + (x-2)(x+3)(x+2) * \frac{x}{x+3} = (x-2)(x+3)(x+2) * \frac{1}{x^2+x-6}

Simplifying the equation, we get:

x(x+3)(x+2) + x(x-2)(x+2) = 1

Expanding and combining like terms, we have:

x^3 + 5x^2 + 4x = 1

Rearranging the equation, we get:

x^3 + 5x^2 + 4x - 1 = 0

This is a cubic equation, which can be solved using various methods such as factoring, synthetic division, or numerical methods like Newton's method.

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When evaluating a one-sided 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 one-sided 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 one-sided 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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