Solve the following modulus equation: |x-1| = |x| +1
2 Answers
embizze | High School Teacher | (Level 2) Educator Emeritus
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`
justaguide | College Teacher | (Level 2) Distinguished Educator
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]