#-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.
