Question

Solve each rational equation. State the restrictions.
2𝑥+3/𝑥−1= 3/𝑥+ 2

Answer 1

To solve this rational equation, we need to eliminate the fractions by finding a common denominator and then solving for x.

First, let's find the common denominator. The denominators in this equation are (x - 1) and (x + 2), so the common denominator is (x - 1)(x + 2).

Next, let's multiply both sides of the equation by the common denominator:
(x - 1)(x + 2) * (2𝑥+3)/(𝑥−1) = (x - 1)(x + 2) * 3/(𝑥+ 2)

Simplifying the equation:
2𝑥 + 3 = 3(x - 1)

Expanding and simplifying further:
2𝑥 + 3 = 3x - 3

Now, let's isolate the variables by bringing all the x terms to one side and the constant terms to the other side:
2𝑥 - 3𝑥 = -3 - 3
-x = -6

Finally, let's solve for x by multiplying both sides of the equation by -1:
x = 6

However, before we can state the restrictions, we need to check if there are any values of x that would make the denominators (x - 1) or (x + 2) equal to zero. If any of these conditions are true, those values of x would be restrictions.

Checking the first denominator:
x - 1 = 0
x = 1

Checking the second denominator:
x + 2 = 0
x = -2

Thus, the restrictions are x ≠ 1 and x ≠ -2.