Question #21967

/x-5/=/x/ find (x)

Expert's answer

Consider three cases of x.

1) x in (-infinity, 0)

Then x-5<0 and x<0, whence

|x-5| =-(x-5)=-x+5

|x| = -x

and so we get the following equation

-x+5=-x

5=0

which is impossible. 2) x in [0,5]

Then x-5<=0 and x>=0, whence

|x-5| = -x+5

|x| = x

and so we get the following equation

-x+5=x

5=2x

x=2.5

3) x in (5, +infinity)

Then x-5>0 and x>0, whence

|x-5| = x-5

|x| = x

and so we get the following equation

x-5=x

-5=0

which is again impossible.

So we obtain a unique solution: x=2.5

1) x in (-infinity, 0)

Then x-5<0 and x<0, whence

|x-5| =-(x-5)=-x+5

|x| = -x

and so we get the following equation

-x+5=-x

5=0

which is impossible. 2) x in [0,5]

Then x-5<=0 and x>=0, whence

|x-5| = -x+5

|x| = x

and so we get the following equation

-x+5=x

5=2x

x=2.5

3) x in (5, +infinity)

Then x-5>0 and x>0, whence

|x-5| = x-5

|x| = x

and so we get the following equation

x-5=x

-5=0

which is again impossible.

So we obtain a unique solution: x=2.5

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