# Absolute value equations with extraneous solutions

#### pempem

I'm a little confused about solving absolute value equations and why sometimes solutions don't seem to make sense.

Take a look at case (ii) on this website:
http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_AbsoluteValueEquations.xml

I understand the process of solving the equation, and I understand how they arrive at the two solutions. The only problem is, 8/3 is not actually a solution! You can see this by the simple fact that the left side of the equation has to be positive (since the entire left side is inside the absolute value "brackets"). Since 8/3 - 5 is a negative number, it can't be equal to the left side! When you plug in 8/3 you get 7/3 = - 7/3 which does not make sense.

The website claims 8/3 is a solution, but it certainly doesn't seem like it is. Can someone explain, in a mathematical sense, why this discrepancy comes about? Is there any way to know that an answer is extraneous or should one always check solutions to absolute value equations to make sure they are indeed solutions?

Thanks!

#### HallsofIvy

Homework Helper
You are completely correct that 8/3 is NOT a solution to |2x- 3|= x- 5.

On the right 2(8/3- 3= 16/3- 9/3= 7/3 while on the right 8/3- 5= 8/3- 15/3= -7/3.

Unfortunately, there are a number of "algebra" and "mathematics" sites like this one that are full of errors.

#### pempem

Good, I thought I was going crazy haha

In that case, what about the second part of my question: why does this discrepancy happen? Is there any way to know that an answer is extraneous or should one always check solutions to absolute value equations to make sure they are indeed solutions?

#### Thecla

According to Wolfram Alpha , x=-2 is also not a solution. This equation has no solutions .

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