Question

X-6/x+1-2+x/1-x=6/x^2-1​

Answer 1

Ответ:

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Answer 2

To solve the given equation, we first need to find a common denominator for the fractions involved. The common denominator for the fractions in this equation is (x + 1)(x - 1), which is the same as the denominator on the right side (6 / (x^2 - 1)).

Let's rewrite the equation with the common denominator:

(x * (x - 1) / (x + 1)(x - 1)) - 2 + (x * (x + 1) / (x + 1)(x - 1)) = 6 / (x^2 - 1)

Now, let's combine the fractions:

[(x * (x - 1) - 2(x + 1) + x * (x + 1))] / (x + 1)(x - 1) = 6 / (x^2 - 1)

Next, simplify the equation:

[(x^2 - x - 2) + (x^2 + x)] / (x^2 - 1) = 6 / (x^2 - 1)

Now, combine like terms in the numerator:

(2x^2 - 2) / (x^2 - 1) = 6 / (x^2 - 1)

Since the denominators on both sides are the same (x^2 - 1), we can eliminate them:

2x^2 - 2 = 6

Now, let's isolate x:

2x^2 = 6 + 2

2x^2 = 8

x^2 = 8 / 2

x^2 = 4

Now, take the square root of both sides to find the possible values of x:

x = ±√4

x = ±2

Therefore, the solutions to the equation are x = 2 and x = -2. However, we need to check these solutions in the original equation since some values may lead to undefined terms in the denominator.

Let's verify the solutions:

For x = 2:

Left side: (x * (x - 1) / (x + 1)(x - 1)) - 2 + (x * (x + 1) / (x + 1)(x - 1)) = (2 * (2 - 1) / (2 + 1)(2 - 1)) - 2 + (2 * (2 + 1) / (2 + 1)(2 - 1)) = (2 / 3) - 2 + (6 / 3) = 2/3 - 2 + 2 = 2/3

Right side: 6 / (x^2 - 1) = 6 / (2^2 - 1) = 6 / (4 - 1) = 6 / 3 = 2

Since the left side does not equal the right side, x = 2 is not a valid solution.

For x = -2:

Left side: (x * (x - 1) / (x + 1)(x - 1)) - 2 + (x * (x + 1) / (x + 1)(x - 1)) = (-2 * (-2 - 1) / (-2 + 1)(-2 - 1)) - 2 + (-2 * (-2 + 1) / (-2 + 1)(-2 - 1)) = (6 / -3) - 2 + (-6 / -3) = -2 + 2 = 0

Right side: 6 / (x^2 - 1) = 6 / ((-2)^2 - 1) = 6 / (4 - 1) = 6 / 3 = 2

The left side is not equal to the right side for x = -2 either.

Therefore, the given equation has no valid solutions.