Question

Solve for x (x+1)/(x+3)+1=(x-4)/(x+2) If there is more than one solution, sep If there is no solution, click on "No soli

Answer 1

To solve for x in the equation (x+1)/(x+3)+1=(x-4)/(x+2), combine fractions by finding a common denominator. After simplifying and factoring, we find that x = -1 or x = -20, both valid solutions.

Step-by-step explanation:

To solve for x in the equation (x+1)/(x+3)+1=(x-4)/(x+2), we will start by finding a common denominator for the fractions and then combine them. We will then solve the resulting equation for x. Let's work through the steps:

1. Find a common denominator: The denominators are x+3 and x+2. A common denominator could be the product of these two, which is (x+3)(x+2).
2. Multiply both sides of the equation by the common denominator to eliminate the fractions:
   (x+1)(x+2) + (x+2)(x+3) = (x-4)(x+3)
3. Expand both sides:
   x^2 + 3x + 2x + 2 + x^2 + 5x + 6 = x^2 + 3x - 4x - 12
4. Simplify and combine like terms:
   2x^2 + 10x + 8 = x^2 - x - 12
5. Subtract x^2 and add x + 12 to both sides to move all terms to one side:
   x^2 + 11x + 20 = 0
6. Factor the quadratic equation:
   (x + 1)(x + 20) = 0
7. Solve for x by setting each factor equal to zero:
   x + 1 = 0 or x + 20 = 0
   Therefore, x = -1 or x = -20

However, we must check for extraneous solutions, as the original equation has denominators that could result in division by zero. In this case, x = -1 and x = -20 are both valid solutions because -1 and -20 do not make any denominator zero.