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Solve the equation AND check your solution. Please show all of your work.

5/x^2-7x+12 - 2/x-3 = 5/x-4

User LuisVM
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1 Answer

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First, we need to find the LCD (Least Common Denominator) of the fractions on both sides of the equation. The denominators of the fractions are x^2 - 7x + 12, x - 3, and x - 4.

The factorization of the denominator of the first fraction is:

x^2 - 7x + 12 = (x - 4)(x - 3)

So, the LCD is (x - 4)(x - 3).

We can now rewrite the equation with the LCD:

5/(x - 4)(x - 3) - 2/(x - 3) = 5/(x - 4)

Multiplying both sides by the LCD, we get:

5 - 2(x - 4) = 5(x - 3)

Simplifying:

5 - 2x + 8 = 5x - 15

Collecting like terms:

7x = 28

x = 4

We have found that x = 4 is the solution to the equation.

To check our solution, we need to verify that it does not produce any denominators equal to zero.

The original equation with x = 4 is:

5/4^2 - 7(4) + 12 - 2/4 - 3 = 5/4 - 4

Simplifying:

5/4 - 4 = 5/4 - 4

Therefore, the solution x = 4 is valid.
User Ingris
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