Question

Solve the rational equation 4x/x+5=x/x+4

Answer 1

To solve the rational equation 4x/(x+5) = x/(x+4), find a common denominator and eliminate the fractions. The solutions are x = 0 and x = -11/3.

Step-by-step explanation:

To solve the rational equation 4x/(x+5) = x/(x+4), we first need to find a common denominator. The common denominator is (x+4)(x+5) since both denominators are factors of it. We then multiply both sides of the equation by the common denominator to eliminate the fractions.

1. Expand the equation: 4x(x+4) = x(x+5)
2. Simplify: 4x^2 + 16x = x^2 + 5x
3. Rearrange the equation: 4x^2 + 16x - x^2 - 5x = 0
4. Combine like terms: 3x^2 + 11x = 0
5. Factor out x: x(3x + 11) = 0
6. Set each factor equal to zero and solve for x: x = 0 or 3x + 11 = 0
7. If x = 0, substitute it back into the original equation to check if it's a valid solution
8. If 3x + 11 = 0, solve for x: 3x = -11, x = -11/3
9. Check if x = -11/3 is a valid solution by substituting it back into the original equation

The solutions to the rational equation are x = 0 and x = -11/3.

Learn more about Solving rational equations