Question

how many answers does \[x+\left| x \right|+\left| x -\left| x \right|\right|=2\]have?
a)1 b)2 c)0 d)a lot answers

Answer 1

\[x+\left| x \right|+\left| x-\left| x \right| \right|=2\]

Answer 2

clearly x=0 is not a solution, so x can be either positive or negative

Answer 3

if `x is positive`, then |x| = x :

\[x+\left| x \right|+\left| x -\left| x \right|\right|=2\]
\[x+x+\left| x-x \right|=2\]
\[2x=2\]
\[x = 1\]

Answer 4

if `x is negative`, then |x| = -x :
\[x-x+\left| x-(-x) \right|=2\]
\[0+\left| 2x \right|=2\]
\[2\left| x \right|=2\]
\[-2x=2\]
\[x=-1\]

Answer 5

You can also consider this question graphically.

Adding the terms is the same as superimposing the plot of one term onto the other.