How do you simplify #\frac { \frac { 4} { x + 4} - 3} { 2+ \frac { 1} { x + 4} }#?

1 Answer
Shantelle
Oct 14, 2017

#(-3x-8)/(2x+5)#

Explanation:

There are a few ways you can solve this problem, and this is one way:
First, let's look at the numerator of the entire fraction:
#4/(x+4)-3#

We have to get this whole expression under the same denominator. In this case, our same denominator is #x+4#. First, #-3# literally means #-3/1#. To get the same denominator, we will multiply the #-3/1# by #x+4#, so it looks like this:
#4/(x+4) - (3 * (x+4))/(1 * (x+4))#

When we simplify that we get:
#4/(x+4) - (3x+12)/(x+4)# and now we have a common denominator, we can combine them together:
#(4-(3x+12))/(x+4)#
#(4-3x-12)/(x+4)# (distribute the negative)
#(-3x-8)/(x+4)#

We repeat the same steps for the denominator of the entire fraction:
#2+1/(x+4)#
#(2(x+4))/(x+4) + 1/(x+4)#
#(2x+4)/(x+4) + 1/(x+4)#
#(2x+4+1)/(x+4)# (combine them now that they have the same denominator)
#(2x+5)/(x+4)# (simplify)

So now the entire expression looks like this:
#(4/(x+4) - (3x+12)/(x+4))/((2x+4)/(x+4)-1/(x+4))#.

If we write this into an equation with #-:#, it looks like:
#(-3x-8)/(x+4) -: (2x+5)/(x+4)#

Now, we want to make the expression multiplying instead of dividing, and we do this by flipping one of the expressions:
#(-3x-8)/(x+4) * (x+4)/(2x+5)#

Now, as you can see we can cross out the #(x+4)# because it is on both the numerator and denominator.
The final answer is #(-3x-8)/(2x+5)#.

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