Final answer:
To solve the given rational equation, we need to eliminate the fractions and potentially use the quadratic formula to find the values of x, being careful to adhere to the restriction that x cannot be -1. The solution may yield two possible x values, but one might be extraneous and must be checked against the restrictions.
Step-by-step explanation:
The student's question is about solving the rational equation (3x)/(x+1)=5-(3)/(x+1). The restrictions on the variable x are determined by the denominators in the equation, as x cannot take a value that would make the denominator zero. In this case, x cannot be -1 since it would make the denominator in both fractions equal to zero, which is undefined.
To solve the equation for x, we need to find a common denominator and combine the terms. Then we can use algebraic methods to isolate x and solve for it. Since (3x)/(x+1) already has the denominator (x+1), we can multiply both sides of the equation by (x+1) to eliminate the fractions. After simplifying, we can apply the quadratic formula if necessary to find the values of x, keeping in mind the restriction that x ≠ -1.
If the equation simplifies to a quadratic, we acknowledge that there may be two solutions because of the ± sign in the quadratic formula. However, one of these solutions may be extraneous and not make sense in the context of the problem, especially if it violates the initial restrictions on the variable x. Therefore, we must check our potential solutions against the restrictions.