How do you solve #(2x-1)/(3x-2) <=1#?

2 Answers
May 10, 2015

Solve #(2x-1)/(3x-2)<=1#

Multiply both sides by #3x-2#

#(cancel (3x-2)xx2x-1)/cancel(3x-2)<=1xx(3x-2)# =

#2x-1<=3x-2#

Subtract #3x# from both sides.

#2x-1-3x<=-2#

Add #1# to both sides.

#2x-3x<=-2+1# =

#-x<=-1#

Multiply by #-1#

#x>=1#

May 10, 2015

#(2x-1)/(3x-2) <= 1#

Case 1:
If #3x-2<0#
which implies #color(red)(x<2/3)#
then
#2x-1 >= 3x-2#
( since multiplying by a negative reverses the inequality)
and
#color(red)(x<= 1)#
Combining the Case 1 restrictions on #x#
#color(red)(x<2/3)# and #color(red)(x<=1)#
gives
#color(red)(x<2/3)#

Case2
If #3x-2>0#
which implies #color(blue)(x>2/3)#
then
#2x-1<=3x-2#
#color(blue)(x>=1)#
Combining the Case 2 restrictions on #x#
#color(blue)(x>2/3)# and #color(blue)(x>=1)#
gives
#color(blue)(x>=1)#

Solution
#x<2/3# or #x>=1#