Solving an inequality with absolute values
#2|2x+5|>=4#
#|2x+5|>=2#
#|2x+5|-2>=0#
#2x+5>=0#, #=>#, #x>=-5/2#
We can build a sign chart
#color(white)(aaaa)##x##color(white)(aaaaaaa)##-oo##color(white)(aaaaaaaaaaa)##-5/2##color(white)(aaaaaaaaa)##+oo#########
#color(white)(aaaa)##2x+5##color(white)(aaaaaaaaaaa)##-##color(white)(aaaaaa)##0##color(white)(aaaa)##+#########
#color(white)(aaaa)##|2x+5|##color(white)(aaaaaaa)##-2x-5##color(white)(aaaa)##0##color(white)(aaaa)##2x+5#########
#color(white)(aaaa)##|2x+5|-2##color(white)(aaaa)##-2x-7##color(white)(aaaa)####color(white)(aaaaa)##2x+3#########
Therefore,
In the Interval #(-oo, -5/2)#
#-2x-7>=0#, #<=>#, #2x<=-7#, #>=>#, #x<=-7/2#
In the Interval #( -5/2,+oo)#
#2x+3>=0#, #<=>#, #2x>=-3#, #>=>#, #x>=-3/2#
The solution is
# x in (-oo, -7/2] uu[-3/2, +oo)#
graph{|2x+5|-2 [-7.9, 7.9, -3.95, 3.95]}
