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Solve for x example

Q:  Solve for x:  4(x-2)/(x-3) + 3/x = -3/(x(x-3))

A:  First, I look at all of the denominators… They are (x – 3), x and x*(x-3).  Notice that x and (x – 3) are the only two factors… Therefore, I am going to multiply the whole equation by x*(x – 3) in order to clear out the denominators… Watch like so:

x*(x – 3)  [ 4(x-2)/(x-3) + 3/x = -3/(x(x-3)) ]

Now, distribute the x*(x – 3) to each part:

4(x-2)*x*(x – 3)/(x-3) + 3*x*(x – 3)/x = -3*x*(x – 3)/(x(x-3))

Now, cancel things out:

4(x-2)*x*(x – 3)/(x-3) + 3*x*(x – 3)/x = -3*x*(x – 3)/(x(x-3))

4(x – 2) * x + 3(x – 3) = -3

Distribute again and solve for x:

(4x – 8)*x + 3x – 9 = -3

4x² – 8x + 3x – 9 = -3

4x² – 5x – 9 = -3

Add 3 to both sides:

4x² – 5x – 6 = 0

Now, solve by factoring or by the quadratic equation… I will factor:

(4x + 3)(x – 2) = 0

So, 4x + 3 = 0 or x – 2 = 0

x = -3/4 or x = 2

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