SOLUTION: If |x|+|x-1|=3 then x=?

Algebra.Com
Question 999927: If |x|+|x-1|=3 then x=?
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
Answer. This equation has two solutions: x = 2 and x = -1.

Solution

1.  If x < 0, then |x| = -x and |x-1| = -(x-1), therefore |x| + |x-1| = -x -(x-1) = -2x + 1. 

    Hence,  the equation |x| + |x-1| = 3 takes the form -2x + 1 = 3.

    The last equation has the solution x = -1,  which satisfies the inequality x < 0.

2.  If 0 <= x < 1, then |x| = x and |x-1| = -(x-1), therefore |x| + |x-1| = x -(x-1) = 1. 

    Hence,  the equation |x| + |x-1| = 3 takes the form 1 = 3.

    The last equation has no solution.

3.  If x >= 1, then |x| = x and |x-1| = (x-1), therefore |x| + |x-1| = x + (x-1) = 2x - 1. 

    Hence,  the equation |x| + |x-1| = 3 takes the form 2x - 1 = 3.

    The last equation has the solution x = 2,  which satisfies the inequality x >= 1.


The plot of the function y = |x| + |x-1| is presented in the Figure below.


 
  

   




      Figure. Plot y = |x| + |x-1|

RELATED QUESTIONS

if 3^x - 1 = 8, then... (answered by Alan3354)
If x+1/x =6, then... (answered by fractalier)
If (2x+1)(x-5) --------- = 0,then x=... (answered by stanbon,mangopeeler07,checkley77)
If f( x ) = ( x - 2 )^3 , then f^-1 ( x ) = (answered by tommyt3rd)
If log(x+3) - log(x+1) = 2, then... (answered by Theo)
if 1/3+1/2+1/x=4, then... (answered by edjones,EMStelley,CharlesG2)
If x - 3, then x2 - 2x + 1=... (answered by Maths68)
if 5^2x-3=1 then... (answered by Fombitz)
if {{{3^(x+1)=9^(2x-1)}}} then... (answered by mananth)