Question

(x+2)/(x+3)-(x+1)/(x-1)=4/(x+3)(x-1) ​

Answer 1

(x+2) / (x+3) - (x+1) / (x-1) = 4 / (x+3)(x-1)

ОДЗ: x є (-inf; -3) u (-3; 1) u (1; +inf)

Домножим на (x+3)(x-1) зная, что это произведение двух знаменателей, поэтому равняться нулю оно не может:

(x+2)(x-1) - (x+1)(x+3) = 4

x² - 2x - 2 - x² - 4x - 4 = 4

-6x = 10

x = -10/6

x = -5/3

Answer 2

To solve the equation:

$\frac{{x+2}}{{x+3}} - \frac{{x+1}}{{x-1}} = \frac{4}{{(x+3)(x-1)}}$

First, let's find a common denominator for the fractions on the left side. The common denominator for $(x+3)$ and $(x-1)$ is $(x+3)(x-1)$.

Using this denominator, we can rewrite the equation as:

$\frac{{(x+2)(x-1)}}{{(x+3)(x-1)}} - \frac{{(x+1)(x+3)}}{{(x+3)(x-1)}} = \frac{4}{{(x+3)(x-1)}}$

Expanding the numerators, we get:

$\frac{{x^2 + x - 2}}{{(x+3)(x-1)}} - \frac{{x^2 + 4x + 3}}{{(x+3)(x-1)}} = \frac{4}{{(x+3)(x-1)}}$

Now, combine the fractions on the left side:

$\frac{{x^2 + x - 2 - (x^2 + 4x + 3)}}{{(x+3)(x-1)}} = \frac{4}{{(x+3)(x-1)}}$

Simplify the numerator:

$\frac{{x^2 + x - 2 - x^2 - 4x - 3}}{{(x+3)(x-1)}} = \frac{4}{{(x+3)(x-1)}}$

Combine like terms in the numerator:

$\frac{{-3x - 5}}{{(x+3)(x-1)}} = \frac{4}{{(x+3)(x-1)}}$

Now, we have the same denominator on both sides of the equation. We can eliminate the denominators by multiplying both sides by $(x+3)(x-1)$:

$-3x - 5 = 4$

Simplify:

$-3x = 4 + 5$

$-3x = 9$

Divide both sides by -3:

$x = -3$

So, the solution to the equation is $x = -3$.