How do you solve #-7/(x+5)<=-8/(x+6)#?

1 Answer
Mar 18, 2018

Answer:

#{x < -6} cup {-5 < x <= 2} #

Explanation:

#-7/(x+5)<=-8/(x+6)#

Let's multiply both sides by #-1#. Since we are multiplying/dividing by a negative value, we must flip the direction of the inequality:

#7/(x+5) >=8/(x+6)#

Multiply both sides by #(x+5)#:

#7 >= (8(x+5))/(x+6)#

Multiply both sides by #(x+6)#:

#7(x+6) >= 8(x+5)#

Distribute #7# into #x+6# and distribute #8# into #x+5#:

#7x+42 >=8x+40#

Subtract #7x# from both sides:

#42 >= x+40#

Subtract #40# from both sides:

#2 >= x#

Hence:

#=>color(blue)(x<=2)#

Now we have to assess the actual inequality for values that #x# cannot be due to the rational terms being undefined.

Looking at the denominator #x+5#, we see #x \ne -5#.

Looking at the denominator #x+6#, we see #x \ne -6#.

So the final result is:

#color(green)({x < -6} cup {-5 < x <= 2} )#

=======================EDIT=======================
Should mention what Douglas mentioned in the comments to make this result clearer.

After we have determined that #x\ne-5# and #x\ne-6#, you should investigate the regions between this "critical" points to determine if that region is included in the final solution. In this case, the example Douglas gives of #-5.5# in the comments suffices to show that the region #-6<x<-5# is not a part of the solution.