Question

Show that there is real number x so that |x−1| = |x−2|.

Answer 1

Function zeros:

x - 1 = 0; x - 2 = 0.

x = 1; x = 2.

Сases:

a) x $\ge$ 2

x - 1 = x - 2

x - x = 1 - 2

0x = -1. Not possible. No solutions.

b) x $\ge$ 1 and x < 2

$\begin{cases} x \in[1,2) \\ x - 1 = -(x-2) \end{cases}$

$\begin{cases} x \in[1,2) \\ x - 1 = -x+2 \end{cases}$

$\begin{cases} x \in[1,2) \\ 2x = 3 \end{cases}$

$\begin{cases} x \in[1,2) \\ x = 1.5 \end{cases}$

x = 1.5 - solution of this equation.

c) x < 1

-(x - 1) = - (x - 2)

-x + 1 = -x +2

-x + x = 2 -1

-0x = 1. Not possible. No solutions.

Answer: 1.5 - one solutions.