Help in absolute value question?!

Solve $| 2 x - 1 | < | x |$. I understand that the correct answer is $\frac{1}{3} \le x \le 1$, however I do not understand that when you apply: $2 x - 1 < 0$ and $x < 0$, the answer comes out as $x \ge 1$, which is not the answer. Please explain?

Jul 29, 2018

Below

Explanation:

Draw the functions $y = \left\mid 2 x - 1 \right\mid$ and $y = \left\mid x \right\mid$ 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:

• $y = x$
• $y = 2 x - 1$
• $y = - \left(2 x - 1\right)$
• $y = - x$

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

Finding the 2 points,

$y = x$ and $y = 2 x - 1$

$x = 2 x - 1$
$x = 1$

$y = x$ and $y = - \left(2 x - 1\right)$

$x = - \left(2 x - 1\right)$
$x = 1 - 2 x$
$3 x = 1$
$x = \frac{1}{3}$

Therefore, the line $y = x$ crosses $y = \left\mid 2 x - 1 \right\mid$ at $x = 1$ and $x = \frac{1}{3}$. Now, to find where the $y = \left\mid 2 x - 1 \right\mid$ is less than $y = \left\mid x \right\mid$, we need to look at the graph. When $y = \left\mid 2 x - 1 \right\mid$ is between $x = \frac{1}{3}$ and $x = 1$, the function is less than $y = \left\mid x \right\mid$ because it is BELOW $y = \left\mid x \right\mid$

Hence, the answer is $\frac{1}{3} < x < 1$