# How do you solve the rational equation #1 / (x+1) = (x-1)/ x + 5/x#?

First, recognize that the fractions on the right hand side can be added since they have the same denominator.

Now, cross multiply.

You will have to FOIL on the right hand side.

From here, you could find the roots by using the quadratic formula, completing the square, or simply by factoring and recognizing this is a perfect square trinomial.

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To solve the rational equation 1 / (x+1) = (x-1)/ x + 5/x, we can start by finding a common denominator for the fractions on both sides. The common denominator in this case is x(x+1).

Multiplying both sides of the equation by x(x+1), we get:

x(x+1) * (1 / (x+1)) = x(x+1) * ((x-1)/ x + 5/x)

Simplifying, we have:

x = (x-1)(x(x+1)) / x + 5(x(x+1)) / x

Expanding and simplifying further, we get:

x = (x^2 - x)(x+1) / x + (5x^2 + 5x) / x

Next, we can distribute and combine like terms:

x = (x^3 + x^2 - x^2 - x) / x + (5x^2 + 5x) / x

Simplifying, we have:

x = (x^3 - x) / x + (5x^2 + 5x) / x

Combining the fractions, we get:

x = (x^3 - x + 5x^2 + 5x) / x

Simplifying further, we have:

x = (x^3 + 5x^2 + 4x) / x

Dividing both sides by x, we get:

1 = x^2 + 5x + 4

Rearranging the equation, we have:

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

Simplifying, we get:

x^2 + 5x + 3 = 0

Now, we can solve this quadratic equation by factoring, completing the square, or using the quadratic formula. In this case, the equation can be factored as:

(x + 3)(x + 1) = 0

Setting each factor equal to zero, we have:

x + 3 = 0 or x + 1 = 0

Solving for x, we get:

x = -3 or x = -1

Therefore, the solutions to the rational equation 1 / (x+1) = (x-1)/ x + 5/x are x = -3 and x = -1.

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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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