How do you simplify #\frac { \frac { 9} { x } - 3} { \frac { 1} { 3} - \frac { 1} { x } }#?

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Answer 1 · 2016-11-30T05:45:05

#-9#

You can't have fractions within a fraction. First, you must combine them in the numerator and denominator by finding common factors:

#(9/x-3)/(1/3-1/x)#

#(9/x-3*x/x)/(1/3*x/x-1/x*3/3)#

#=(9/x-(3x)/x)/(x/(3x)-3/(3x)#

#=((9-3x)/x)/((x-3)/(3x))#

When you divide a fraction by another fraction, it is the same thing as multiplying by the second's reciprocal.

#(-3x+9)/x*(3x)/(x-3)#

You can take out a common factorof -3 in the first numerator and cancel out the #x#s of the left denominator and right numerator.

#(-3x+9)/color(red)x*(3color(red)x)/(x-3)#

#-3cancel((x-3))*(3)/cancel((x-3))#

#=-9#