Question

-x+5 < 1/x+3
The solution manual is my book shows this

-x+5 - 1/x+3 < 0

-x^2 + 2x + 14/x + 3 < 0

(x+2.873)(x-4.873)/x+3 < 0

I understand everything except how the (-) sign from the -x^2 in the
-x^2 + 2x + 14/x + 3 < 0
step disappeared in the
(x+2.873)(x-4.873)/x+3 < 0 step??

Answer 1

That is a problem. They clearly multiplied the left side by -1, so the right side should have been > 0

(-x^2 + 2x + 14)/(x + 3) < 0 , those brackets are essential
(x^2 - 2x - 14)/(x+3) > 0
(x+2.873)(x-4.873)/(x+3) > 0

the critical values are -3, which we can't have
x = -2.873 and x = 4.873

so place those values on the x-axis and consider the 4 sections
x < -3 , say x = -5 in the orginal
LS = 10 , RS = 1/8 , LS < RS is false

x between -3 and -2.873 , say x = -2.9
LS = 7.9, RS = 1/.1= 10 , that is true

x between -2.873 and 4.873 , say x = 0
LS = 5, RS = 1/3 , is 5 < 1/3 , NO

x > 4.873 , say x = 6
RS = -1 , RS = 1/9 , LS < RS

so -3 < x < -2.873 OR x > 4.873

check my algebra