#6x - 5 < 6/x#
We need to get all the #x#s on one side
multiply both sides by #x#
#x(6x - 5) < 6#
distribute
#6x^2 - 5x < 6#
Looks like a quadratic. Let's bring the #6# over and set the inequality to #0#
#6x^2 - 5x - 6 < 0#
Now we need to factor #6x^2 - 5x - 6#
# color(white)(..) + 36#
# color(white)(..) xx 5#
. . . . . . . . .
# color(white)(..) 1 xx 36#
# color(white)(..) 2 xx 18#
# color(white)(..) 3 xx 12#
# color(white)(..) color(orange)(4) xx color(white)(0)color(orange)(9)color(white)(0)# #=> color(orange)(-9)+color(orange)(4)#
Since the leading coefficient of the expression isn't #1#, we need to factor by grouping
#(6x^2 + color(orange)(4)x) + (color(orange)(-9)x - 6)#
#2x(3x + 2) + -3(3x + 2)#
#(2x - 3)(3x + 2)#
So now we have
#(2x - 3)(3x + 2) < 0#
#* * * * * * * * * * * * * * * * * *#
Solve for #x# in #(2x - 3)#:
# 2x - 3 < 0 #
# 2x < 3 #
#x < 3/2#
#* * * * * * * * * * * * * * * * * *#
Solve for #x# in #(3x + 2)#:
#3x + 2 < 0#
#3x < -2#
#x < -2/3#
So #-2/3 < x < 3/2#
