Question 8495: It's been awhile since I've done absolute value and I'm having trouble with |2x-1|-|x+5|=3.
Answer by khwang(438)   (Show Source):
You can put this solution on YOUR website! |2x-1|-|x+5|=3
The root of 2x-1 =0 is x= 1/2
and the root of x+5 =0 is x= -5.
As
-oo -5 1/2 +oo
-----------*---------------*---------------
Note, -5 < 1/2 and these two roots divide the real line into 3 parts
(-oo,-5) , [-5,1/2] and (1/2,+oo).
Case (i) if x is in (-oo,-5) , since x+5 < 0 and 2x-1 <0
then |2x-1| = -(2x-1) = -2x+1, and |x+5| = -x-5.
So,|2x-1|-|x+5|=3 is equivalent to -2x+1 -x-5 =3
and we get -3x = 7, x = -7/3 .
But -7/3> -5 , so it is an invalid answer.
Case (ii) if x is in [-5,1/2] , since x+5 >= 0 and 2x-1 <= 0
then |2x-1| = -(2x-1) = -2x+1, and |x+5| = x+5.
So,|2x-1|-|x+5|=3 is equivalent to -2x+1 +x+5 =3
and we get -x = -3, x = 3.
But -3 is not in [-5,1/2] , so it is an invalid answer.
Case (iii) if x is in (1/2,+oo) , since x+5 < 0 and 2x-1 > 0
then |2x-1| = 2x-1, and |x+5| = -x-5.
So,|2x-1|-|x+5|=3 is equivalent to 2x-1 -x-5 =3
and we get x = 9.
But 9 is in (1/2,+oo) , so it is a valid answer.
We claim that there is only one solution x = 9.
If you have trouble understanding ,try to review the definition about
the absolute value, think carfully and draw
a diagram on the real line. Sorry ,I won't give you further explanations.
Kenny
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