Question 278468
{{{((x-4)/(x-1) - x/(x-3))/(3+ 12/(x-3))}}}
The LCD of all the "little" fractions is the right approach. The LCD here is:
(x-1)(x-3)
Multiplying the numerator and denominator of the "big" fraction by the LCD we get:
{{{(((x-4)/(x-1) - x/(x-3))/(3+ 12/(x-3)))((x-1)(x-3)/(x-1)(x-3))}}}
To multiply this we need to use the Distributive Property on both the numerator and denominator:
{{{((x-1)(x-3)((x-4)/(x-1)) - (x-1)(x-3)(x/(x-3)))/((x-1)(x-3)3+ (x-1)(x-3)(12/(x-3))))}}}
Now we can start canceling:
{{{(cross((x-1))(x-3)((x-4)/cross((x-1))) - (x-1)cross((x-3))(x/cross((x-3))))/((x-1)(x-3)3+ (x-1)cross((x-3))(12/cross((x-3)))))}}}
leaving:
{{{((x-3)(x-4) - (x-1)(x))/((x-1)(x-3)3+ (x-1)(12))}}}
Now we can finally multiply:
{{{((x^2 - 7x +12) - (x^2 - x))/((3x^2 -12x + 9) + (12x - 12))}}}
Combining like terms we get:
{{{(-6x + 12)/(3x^2 - 3)}}}
Now we can reduce the fraction. To reduce fractions you cancel factors which are common to the numerator and denominator. So we need factors:
{{{(-6(x -2))/(3(x^2 - 1))}}}
{{{(-6(x -2))/(3(x+1)(x - 1))}}}
A factor of 3 will cancel:
{{{(-2(x -2))/((x+1)(x - 1))}}}
No more reducing is possible so we'll multiply out the top and bottom:
{{{(-2x+4)/(x^2 -1)}}}