(x-2)/(x+2)+6(x-2/x-6)=1

Jasonbradley

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Jun 17, 2024

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Solving the Equation: (x-2)/(x+2) + 6(x-2)/(x-6) = 1

This article will guide you through the steps of solving the equation:

(x-2)/(x+2) + 6(x-2)/(x-6) = 1

Step 1: Finding the Least Common Multiple (LCM)

The first step is to find the LCM of the denominators: (x+2) and (x-6). The LCM is simply the product of these two factors:

LCM = (x+2)(x-6)

Step 2: Multiplying Each Term by the LCM

Multiply each term of the equation by the LCM:

(x+2)(x-6) * [(x-2)/(x+2)] + (x+2)(x-6) * [6(x-2)/(x-6)] = (x+2)(x-6) * 1

This simplifies to:

(x-6)(x-2) + 6(x+2)(x-2) = (x+2)(x-6)

Step 3: Expanding and Simplifying

Expand the products on both sides of the equation:

(x² - 8x + 12) + 6(x² - 4) = (x² - 4x - 12)

Further simplification gives:

x² - 8x + 12 + 6x² - 24 = x² - 4x - 12

Step 4: Combining Like Terms

Combine the terms with the same power of x on both sides of the equation:

6x² - 8x - 12 = -4x - 12

Step 5: Solving for x

Rearrange the equation to isolate the x terms on one side:

6x² - 4x = 0

Factor out a 2x:

2x(3x - 2) = 0

This gives us two possible solutions:

2x = 0 or 3x - 2 = 0

Solving for x:

x = 0 or x = 2/3

Step 6: Checking for Extraneous Solutions

It's crucial to check if these solutions are valid by plugging them back into the original equation. If any solution results in a division by zero, it's an extraneous solution and must be discarded.

Checking x = 0:

(0-2)/(0+2) + 6(0-2)/(0-6) = 1

This simplifies to:

-1 + 2 = 1

1 = 1

This solution is valid.

Checking x = 2/3:

(2/3-2)/(2/3+2) + 6(2/3-2)/(2/3-6) = 1

This simplifies to:

-4/8 + 6(-4)/-16 = 1

-1/2 + 3/2 = 1

1 = 1

This solution is also valid.

Conclusion

Therefore, the solutions to the equation (x-2)/(x+2) + 6(x-2)/(x-6) = 1 are:

x = 0 and x = 2/3.