How do you solve #|1- 6x | - 9< - 6#?

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Answer 1 · 2018-05-09T09:03:35

The solution is # x in (-1/3, 2/3)#

This is an inequality with absolute values

#|1-6x|-9<-6#

#|1-6x| < 3#

Therefore,

#{(1-6x<3),(-1+6x<3):}#

#<=>#, #{(6x> -2),(6x<4):}#

#<=>#, #{(x> --1/3),(x<2/3):}#

The solution is # x in (-1/3, 2/3)#

graph{|1-6x|-3 [-6.244, 6.24, -3.12, 3.12]}

Answer 2 · 2018-05-09T09:15:57

#color(blue)((-1/3,2/3)#

#|1-6x|-9<-6#

Add 9 to both sides of the inequality:

#|1-6x| <3#

By the definition of absolute value:

#|a| = acolor(white)(8888.8)bb("if and only if")color(white)(888) a>=0#

#|a| = -acolor(white)(888)bb("if and only if")color(white)(888) a < 0#

From this definition we notice that we have to solve both:

#1-6x <3 and -(1-6x) <3#

#:.#

#1-6x < 3#

#6x> -2#

#x > -1/3#

#-(1-6x) <3#

#-1+6x < 3#

#6x < 4#

#x < 2/3#

The solution range in interval notation is:

#(-1/3,2/3)#