Question

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?

Answer 1

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`