How do you solve the equation abs(2x-1)=x+1?

Aug 15, 2017

See below.

Explanation:

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

now assuming $2 x - 1 \ne 0$ we have

$\frac{\left\mid 2 x - 1 \right\mid}{2 x - 1} = \frac{1}{2} + \frac{3}{2} \frac{1}{2 x - 1}$ so

$2 \frac{\left\mid 2 x - 1 \right\mid}{2 x - 1} = 1 + \frac{3}{2 x - 1}$

but $\frac{\left\mid 2 x - 1 \right\mid}{2 x - 1} = \pm 1$ so we have two options

$\left\{\begin{matrix}- 2 = 1 + \frac{3}{2 x - 1} \to - 3 = \frac{3}{2 x - 1} \to 2 x - 1 = - 1 \to x = 0 \\ 2 = 1 + \frac{3}{2 x - 1} \to 1 = \frac{3}{2 x - 1} \to 2 x - 1 = 3 \to x = 2\end{matrix}\right.$

so the solutions are

$x = \left\{0 , 2\right\}$