# Find the values of x in?  |x-1|-|2x-5|=2x

Jul 24, 2018

There is only one solution $x = - 4$.

#### Explanation:

There are $2$ points to consider

$x - 1 = 0$, $\implies$, $x = 1$

$2 x - 5 = 0$, $\implies$, $x = \frac{5}{2}$

There are $3$ intervals to consider

${I}_{1} = \left(- \infty , 1\right)$

${I}_{2} = \left(1 , \frac{5}{2}\right)$

${I}_{3} = \left(\frac{5}{2} , + \infty\right)$

In the first interval ${I}_{1}$

$- x + 1 + 2 x - 5 - 2 x = 0$

$x = - 4$

This solution $\in {I}_{1}$

In the second interval ${I}_{2}$

$x - 1 + 2 x - 5 - 2 x = 0$

$x = 6$

This solution does not $\notin {I}_{2}$

In the third interval ${I}_{3}$

$x - 1 - 2 x + 5 - 2 x = 0$

$3 x = 4$

$x = \frac{4}{3}$

This solution does not $\notin {I}_{3}$

There is only one solution $x = - 4$

graph{|x-1|-|2x-5|-2x [-18.02, 18.02, -9.01, 9.02]}