You cannot cross multiply when you have an inequality
The inequality is
#|5/(2x-1)|>=|1/(x-2)|#
#<=>#, #5/|2x-1|>=1/|(x-2)|#
#<=>#, #5/|2x-1|-1/|(x-2)|>=0#
There are #2# points to consider
#2x-1=0#, #=>#, #x=1/2# and
#x-2=0#, #=>#, #x=2#
There are #3# intervals to consider
#I_1=(-oo, 1/2)# and #I_2=(1/2, 2)# and #I_3=(2,+oo)#
In the first interval #I_1#, we have
#5/(-2x+1)-1/(-x+2)>=0#
#(5(2-x)-1(1-2x))/((1-2x)(2-x))>=0#
#(10-5x-1+2x)/((1-2x)(2-x))>=0#
#(9-3x)/((1-2x)(2-x))>=0#
#(3(3-x))/((1-2x)(2-x))>=0#
Let #f(x)=(3(3-x))/((1-2x)(2-x))#
Solving this equation with a sign chart
#color(white)(aaaa)##x##color(white)(aaaa)##-oo##color(white)(aaaaaaa)##1/2##color(white)(aaaaaa)##2##color(white)(aaaa)##3##color(white)(aaaa)##+oo#
#color(white)(aaaa)##1-2x##color(white)(aaaa)##+##color(white)(aaaa)##||##color(white)(aaa)##-##color(white)(aaa)##-##color(white)(aaa)##-#
#color(white)(aaaa)##2-x##color(white)(aaaa)##+##color(white)(aaaa)####color(white)(aaaa)##+##color(white)(aa)##||##color(white)(a)##-##color(white)(aaa)##-#
#color(white)(aaaa)##3-x##color(white)(aaaa)##+##color(white)(aaaa)####color(white)(aaaa)##+##color(white)(aaaa)##+##color(white)(a)##0##color(white)(a)##-#
#color(white)(aaaa)##f(x)##color(white)(aaaaa)##+##color(white)(aaaa)##||##color(white)(aaaa)##-##color(white)(a)##||##color(white)(a)##+##color(white)(a)##0##color(white)(a)##-#
Therefore,
In the interval #I_1#, #f(x)>=0#,when #x in (-oo,1/2)#
In the second interval #I_2=(1/2,2)#
#5/(2x-1)-1/(-x+2)>=0#
#<=>#, #(5(2-x)-1(2x-1))/((2x-1)(2-x))>=0#
#<=>#, #(10-5x-2x+1)/((2x-1)(2-x))>=0#
#<=>#, #(11-7x)/((2x-1)(2-x))>=0#
Let #g(x)=(11-7x)/((2x-1)(2-x))#
Solving this inequality with a sign chart,
#g(x)>=0# when #x in (1/2,11/7]#
In the third interval #I_2=(2, +oo)#
#5/(2x-1)-1/(x-2)>=0#
#<=>#, #(5(x-2)-1(2x-1))/((2x-1)(x-2))>=0#
#<=>#, #((5x-10-2x+1))/((2x-1)(x-2))>=0#
#<=>#, #((3x-9))/((2x-1)(x-2))>=0#
#<=>#, #(3(x-3))/((2x-1)(x-2))>=0#
#h(x)=(3(x-3))/((2x-1)(x-2))#
Solving this inequality with a sign chart,
#h(x)>=0# when #x in [3,+oo)#
graph{5/(|2x-1|)-1/(|x-2|) [-14.24, 14.24, -7.12, 7.12]}