Question 999927
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<U>Answer</U>. This equation has two solutions: x = 2 and x = -1.

<U>Solution</U>

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

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

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

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

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

&nbsp;&nbsp;&nbsp;&nbsp;The last equation has no solution.

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

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

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


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

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{{{graph( 330, 330, -5.5, 5.5, -5.5, 5.5,
          abs(x) + abs(x-1)
)}}}


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;<B>Figure</B>. Plot y = |x| + |x-1|

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