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Solving the Equation: 10/(x-5)(x+1) + x/(x+1) = 3/(x-5)
This equation may look complex, but don't worry, we can break it down step by step to solve for x.
Step 1: Simplify the Equation
Let's start by simplifying the equation:
10/(x-5)(x+1) + x/(x+1) = 3/(x-5)
We can start by combining the fractions on the left side of the equation:
(10 + x) / (x-5)(x+1) = 3/(x-5)
Step 2: Cross-Multiply
Now, let's cross-multiply to eliminate the fractions:
(10 + x)(x-5) = 3(x+1)
Step 3: Expand and Simplify
Expanding the equation, we get:
10x - 50 + x^2 - 5x = 3x + 3
Combine like terms:
x^2 - 2x - 53 = 0
Step 4: Factor the Quadratic
Now, let's factor the quadratic equation:
(x - 7)(x + 7) = 0
Step 5: Solve for x
This gives us two possible solutions for x:
x - 7 = 0 => x = 7
x + 7 = 0 => x = -7
Therefore, the solutions to the equation are x = 7 and x = -7.
Conclusion
In this article, we showed how to solve the equation 10/(x-5)(x+1) + x/(x+1) = 3/(x-5) step by step. By simplifying, cross-multiplying, expanding, and factoring, we arrived at the solutions x = 7 and x = -7.
