Solve each rational equation. State the restrictions
6 / 𝑥−1 + 2𝑥 / 𝑥−2 = 2
To solve the given rational equation:
1. Determine the common denominator and combine the fractions on the left side of the equation.
2. Set the resulting fraction equal to the constant on the right side of the equation.
3. Solve the resulting linear equation for the variable.
4. Check the solutions for any restrictions on the variable by ensuring the denominators do not equal zero.
Let's follow these steps:
1. Determine the common denominator and combine the fractions:
The common denominator is (x - 1)(x - 2). Multiply the first fraction by (x - 2) and the second fraction by (x - 1) to obtain a common denominator:
[(6(x - 2)) + (2x(x - 1))]/[(x - 1)(x - 2)] = 2
2. Set the resulting fraction equal to the constant 2:
[6(x - 2) + 2x(x - 1)]/[(x - 1)(x - 2)] = 2
3. Solve the resulting linear equation for the variable:
6(x - 2) + 2x(x - 1) = 2[(x - 1)(x - 2)]
6x - 12 + 2x^2 - 2x = 2(x^2 - 3x + 2)
6x - 12 + 2x^2 - 2x = 2x^2 - 6x + 4
Combine like terms:
2x^2 - 6x + 4 - 2x^2 + 2x - 2x = 12 - 4
Simplify:
0 = 8
Since 0 = 8 is not a true statement, this equation has no solution.
4. Check the solutions for any restrictions on the variable:
In this case, there is no variable remaining in the denominator, so there are no restrictions on the variable.
Therefore, the rational equation 6/(x - 1) + 2x/(x - 2) = 2 has no solution.