## Solve the following modulus equation: |x-1| = |x| +1

Here's another method:

`|x-1|=|x|+1`

`|x-1|^2=(|x|+1)^2` Square both sides. This may introduce extraneous solutions, so you must check any solutions in the original equation.

`x^2-2x+1=x^2+2|x|+1` Note that we do not need absolute value on the left hand side, as squaring results in a nonnegative answer.

`-2x=2|x|`

`-x=|x|` or `|x|=-x` . By definition `:`

|x|= x if x>0, 0 if x=0, and -x if x<0.

Therefore the solution is `x<= 0`

The modulus of a number x is a value equal to -x if x < 0 and equal to x if x >= 0.

We have to solve |x-1| = |x| +1

|x-1| = |x| +1

If x - 1 > 0 => x > 1 or x > 0

=> x - 1 = x + 1,

=> -1 = 1, not possible

If x - 1 =< 0 , x can be greater than or equal to 0 or less than 0

=> 1 - x = x + 1 => x = 0

1 - x = -x + 1 => 1 = 1, always true. The given equation is true for all values of x<0

**The required solution of the equation is [-inf. , 0]**

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