How to solve |x|+ x = 0 ?

Question

-firstly, I isolated for x

|x| = -x

 

then I considered 

x<0

x -|x| = 0

x-x=0

0=0

does this mean x can be any number?

 

then I considered if x≥ 0

x+|x|= 0

x+x=0

2x=0

x=0

 

does that mean that my final answer is that x can be any value? how do i combine the ranges of my two answers?

 

Answer 1

The rule for absolute value is:

  |N|  =   N if N>0

             -N if N<0

              0 if N=0

 

 

Zero is a solution to the equation.

 

if x>0, Then |x|=x

 

So |x|+x = 0

      x + x = 0

       2x = 0

      x=0

which contradicts the fact that x>0

 

if x<0, then |x|=-x

 So |x|+x = 0

      -x + x = 0

       0 = 0

So any negative number satisfies the equation.

 

The solution set is any NON positive real number, negative infinity to zero inclusive