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Solving the Equation: 1/(x - 3) + 2/(x - 2) = 8/x
In this article, we will solve the equation 1/(x - 3) + 2/(x - 2) = 8/x, given that x is not equal to 0, 2, or 3.
Step 1: Find the Least Common Multiple (LCM)
First, let's find the least common multiple (LCM) of the denominators, which are x - 3, x - 2, and x.
The LCM is x(x - 2)(x - 3).
Step 2: Multiply Both Sides by the LCM
Now, let's multiply both sides of the equation by the LCM:
x(x - 2)(x - 3) [1/(x - 3) + 2/(x - 2) = 8/x]
This gives us:
x - 2 + 2x - 4 = 8(x - 2)(x - 3)/x
Step 3: Simplify the Equation
Next, let's simplify the equation by expanding the right-hand side:
x - 2 + 2x - 4 = 8x - 16 - 24 + 48/x
Step 4: Combine Like Terms
Now, let's combine like terms:
3x - 6 = 8x - 40 + 48/x
Step 5: Cross-Multiply
Let's cross-multiply to eliminate the fraction:
3x - 6 = 8x - 40 + 48/x
X(3x - 6) = x(8x - 40 + 48)
Step 6: Solve for x
Now, let's solve for x:
3x^2 - 6x = 8x^2 - 40x + 48
Subtract 8x^2 from both sides:
-5x^2 + 34x - 48 = 0
Factor the quadratic equation:
-(x - 4)(5x + 12) = 0
This gives us two possible values for x:
x - 4 = 0 => x = 4
5x + 12 = 0 => x = -12/5
Conclusion
Therefore, the solutions to the equation 1/(x - 3) + 2/(x - 2) = 8/x are x = 4 and x = -12/5.
