Solve each rational equation. State the restrictions.
2π₯+3/π₯β1= 3/π₯+ 2
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) * (2π₯+3)/(π₯β1) = (x - 1)(x + 2) * 3/(π₯+ 2)
Simplifying the equation:
2π₯ + 3 = 3(x - 1)
Expanding and simplifying further:
2π₯ + 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:
2π₯ - 3π₯ = -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 β 1 and x β -2.