Help in absolute value question?!

Solve #|2x-1|<|x|#.

I understand that the correct answer is #1/3<=x<=1#, however I do not understand that when you apply:
#2x-1<0# and #x<0#,
the answer comes out as #x>= 1#, which is not the answer.
Please explain?

1 Answer
Lucy
Jul 29, 2018

Below

Explanation:

Draw the functions #y=abs(2x-1)# and #y=absx# on the same graph

graph{(y-abs(2x-1))(y-absx)=0 [-10, 10, -5, 5]}

Now, hopefully you can see that the two functions cross each other at 2 different points.

To find your 2 points, you have to know which lines to use. Going from the right, the equations of each lines are:

  • #y=x#
  • #y=2x-1#
  • #y=-(2x-1)#
  • #y=-x#

Looking at the graph, you can see that the line #y=x# crosses #y=2x-1# and #y=-(2x-1)# at exactly one point on each line

Finding the 2 points,

#y=x# and #y=2x-1#

#x=2x-1#
#x=1#

#y=x# and #y=-(2x-1)#

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

Therefore, the line #y=x# crosses #y=abs(2x-1)# at #x=1# and #x=1/3#. Now, to find where the #y=abs(2x-1)# is less than #y=absx#, we need to look at the graph. When #y=abs(2x-1)# is between #x=1/3# and #x=1#, the function is less than #y=absx# because it is BELOW #y=absx#

Hence, the answer is #1/3 < x < 1#

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