SOLUTION: My problem is Absolute Value Equations- Variables on Both Sides.I don't know how to solve the equation. Here is my problem: |x-2|=3x+1.

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Question 658150: My problem is Absolute Value Equations- Variables on Both Sides.I don't know how to solve the equation. Here is my problem: |x-2|=3x+1.
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Found 2 solutions by chriswen, Alan3354:
Answer by chriswen(106) (Show Source):
You can put this solution on YOUR website!
|x-2|=3x+1
The first stop of finding absolute value functions is by finding the breakpoints. This is when the function changes in normal terms. The break point is basically just the zero of the absolute function.
x-2=0
x=2
The breakpoint is at 2. So when x is greater than 2, |x-2| will behave just like x-2. But, when x-2 is less than 2, |x-2| will behave like -(x-2). SO, to the solving part.
CASE 1: (Assume x is greater or equal to 2)
x-2 = 3x + 1
-3 = 2x
x = -3/2
This doesn't follow with our assumption. We can conclude that x isn't >= 2.
CASE 2: (Assume x is less than 2)
-(x-2) = 3x + 1
-x + 2 = 3x + 1
1 = 4x
x = 1/4
This agrees with our assumption.
After studying both cases we can conclude that x is equal to 1/4.

Answer by Alan3354(69443) (Show Source):
You can put this solution on YOUR website!
|x-2|=3x+1
Case 1:
x-2 = 3x+1
2x = -3
x = -3/2
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Check:
|-3/2 - 2| = 3*(-3/2) + 1
|-7/2| = -9/2 + 1 = -7/2
7/2 = -7/2
That doesn't work, reject it.
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Case 2:
-(x-2) = 3x+1
2-x = 3x+1
4x = 1
x = 1/4
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Check:
|1/4 - 2| = 3*(1/4) + 1
|-7/4 | = 7/4
That's ok
x = 1/4