Solve |2x-3|+|x-1|=|x-2| Find the values of x?

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Answer 1 · 2018-06-24T08:08:09

The solutions are $S={1, 3/2}$

The equation is

$|2x-3|+|x-1|=|x-2|$

There are $3$ points to consider

${(2x-3=0),(x-1=0),(x-2=0):}$

$=>$ , ${(x=3/2),(x=1),(x=2):}$

There are $4$ intervals to consider

${(-oo,1),(1,3/2),(3/2,2),(2,+oo):}$

On the first interval $(-oo,1)$

$-2x+3-x+1=-x+2$

$=>$ , $2x=2$

$=>$ , $x=1$

$x$ fits in this interval and the solution is valid

On the second interval $(1, 3/2)$

$-2x+3+x-1=-x+2$

$=>$ , $0=0$

There is no solution in this interval

On the third interval $( 3/2,2)$

$2x-3+x-1=-x+2$

$=>$ , $4x=6$

$=>$ , $x=6/4=3/2$

$x$ fits in this interval and the solution is valid

On the fourth interval $(2, +oo)$

$2x-3+x-1=x-2$

$=>$ , $2x=2$

$=>$ , $x=1$

$x$ does not fit in this interval.

The solutions are $S={1, 3/2}$