How do you solve #3/(x+4)-1/(x+3)=(x+9)/(x^2+7x+12)# and check for extraneous solutions?

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Answer 1 · 2016-12-30T09:55:08

#x=4#

Given:

#3/(x+4)-1/(x+3) = (x+9)/(x^2+7x+12)#

Multiply both sides by:

#(x+4)(x+3) = x^2+7x+12#

to get:

#3(x+3)-1(x+4) = x+9#

Expanding, this becomes:

#3x+9-x-4 = x+9#

Simplifying:

#2x+5 = x+9#

Subtract #x+5# from both sides to get:

#x = 4#

This solution is valid since #(color(blue)(4)+4) = 8 != 0# and #(color(blue)(4)+3) = 7 != 0#.