# Can you simplify (8x-6)/(-4x+3)(x-2) any further?

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• ... ....8x - 6

=--------------------

..( - 4x + 3)(x - 2)

....2(4x - 3)

=--------------------- cancel out 4x - 3

..- (4x - 3)(x - 2)

......... 2

=- ------------

.......x - 2

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• (8x - 6)/(-4x + 3)(x - 2)

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

= 2/(2 - x)

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• (8x - 6)/(-4x + 3)(x - 2)

= (8x - 6)(x - 2)/(-4x + 3)

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

= -2(x - 2) ... x ≠ 3/4

= 4 - 2x ... x ≠ 3/4

There may be another order-of-operations issue here, but I am taking you at your word. If you meant something else, I will let you deal with it.

A more important point here concerns the removable discontinuity. That is an unfortunate expression, because many interpret it as meaning we can remove it and forget about it. We cannot. The given expression is undefined at x = 3/4. No amount of simplification can change that.

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• Yes you can.

(8x - 6) / (-4x + 3)(x - 2)

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

= -2 / (x - 2).

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• As written:

(8x - 6) / (-4x + 3)(x - 2)

The (x - 2) is multiplied by the previous quotient.  So this is the same as:

(8x - 6)(x - 2) / (-4x + 3)

Now we can factor out the 2 from (8x - 6) and the -1 from the (-4x + 3) to get:

2(4x - 3)(x - 2) / [-(4x - 3)]

the (4x - 3)'s cancel out:

2(x - 2) / (-1)

And we can apply the -1 to the numerator to get:

-2(x - 2)

and simplify:

-2x + 4

---------------

If you meant the (x - 2) to be in the denominator, then the problem should be written as:

(8x - 6) / [(-4x + 3)(x - 2)]

Which then we can do the same factoring as before:

2(4x - 3) / [-(4x - 3)(x - 2)]

Cancel the (4x - 3):

2 / [-(x - 2)]

And apply the -1 to the numerator:

-2 / (x - 2)

It depends on what you really meant.  That's why the proper use of parenthesis is important.

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• (8x-6)/(-4x+3)(x-2) = 2(4x-3)/(-4x+3)(x-2) = -2/(x-2)

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