To solve the rational equation 6/(x-1) + 2x/(x-2) = 2, we need to find the common denominator of the two fractions on the left side. The common denominator is (x-1)(x-2).
Multiplying each term by the common denominator, we get:
6(x-2) + 2x(x-1) = 2(x-1)(x-2)
Expanding and simplifying both sides of the equation:
6x - 12 + 2x^2 - 2x = 2x^2 - 2x - 2
Simplifying further:
8x - 12 = 2x^2 - 2x - 2
Rearranging the equation to set it equal to zero:
2x^2 - 10x + 10 = 0
Factoring out a 2:
2(x^2 - 5x + 5) = 0
Setting each factor equal to zero:
x^2 - 5x + 5 = 0
Using the quadratic formula, we can find the solutions for x:
x = (-(-5) ± √((-5)^2 - 4(1)(5))) / (2(1))
x = (5 ± √(25 - 20)) / 2
x = (5 ± √5) / 2
So the solutions to the rational equation are:
x = (5 + √5) / 2
x = (5 - √5) / 2
However, we need to check if these solutions satisfy the restrictions of the equation. In this case, we have restrictions for the denominators:
For x - 1, x ≠ 1
For x - 2, x ≠ 2
Therefore, the restrictions for the equation 6/(x-1) + 2x/(x-2) = 2 are:
x ≠ 1 and x ≠ 2
Solve each rational equation. State the restrictions
6/𝑥−1+2𝑥/𝑥−2= 2
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