Given #1/2x-3 <= -1/1.5x =.5 <= 1/2x+2#
Simplify by multiplying each expression by #6# to clear the fractions (Remember that you can multiply by any value greater than zero without effecting the orientation of the inequalities).
#3x-9 <= -4x-3 <= 3x+12#
Break this up into two compound inequalities:
#"[1] "3x-9<=-4x-3#
and
#"[2] "-4x-3<=3x+12#
Evaluate Inequality [1]
#3x-9<=-4x-3#
#color(white)("MMMMMM")#Add #(4x+9)# to both sides:
#7x <= 6#
#color(white)("MMMMMM")#Divide by #7# (which does not change the inequality orientation)
#x<=6/7#
Evaluate Inequality [2]
#-4x-3 <= 3x+12#
#color(white)("MMMMMM")#Add #(4x-12)# to both sides
#-15<= 7x#
#color(white)("MMMMMM")#Divide both sides by 7
#(-15/7)<=x#
Combine the Compound Inequalities (with and)
#x<=6/7# and #x>= (-15/7)#
#color(white)("MMMMMM")#Re-write as a single compound statement
#(-15/7) <= x <= 6/7#
