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]

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?

Thanks for your response!

 

[Thecla]

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

 

[CellCoree]

Dear student,

Thank you for bringing up this concern about absolute value equations and extraneous solutions. This is a common confusion that many students have when first learning about absolute value equations, so you are not alone in your confusion. Let's dive into the mathematical explanation behind this discrepancy and how to identify extraneous solutions.

First, it is important to understand that absolute value is a mathematical function that always returns a positive value. In other words, it "absolutizes" a number, making it positive regardless of its original sign. This is why, in the solution process for absolute value equations, we must consider both the positive and negative values of the expression inside the absolute value brackets.

Now, let's take a closer look at the example on the website you provided. The equation is |3x - 5| = 7 and the solution process results in two solutions, x = 4 and x = 8/3. As you correctly pointed out, when we plug in x = 8/3 into the equation, we get 7/3 = -7/3, which is not a true statement. So why does this happen?

The reason is because when we solve for x, we are essentially removing the absolute value brackets and solving for the expression inside it. In this case, we are solving for 3x - 5. When we do this, we end up with two equations: 3x - 5 = 7 and 3x - 5 = -7. These two equations represent the two possible values of 3x - 5 that would give us a positive 7 when we apply the absolute value function.

Therefore, when we plug in x = 8/3 into the equation 3x - 5 = -7, we get a false statement. This is because the expression inside the absolute value brackets must be positive, but in this case, it is negative. This means that x = 8/3 is an extraneous solution, or a solution that does not satisfy the original equation.

To answer your question, yes, it is important to always check solutions to absolute value equations to make sure they are indeed solutions. This is because the nature of absolute value equations can result in extraneous solutions. A good way to check for extraneous solutions is to plug the solutions back into the original equation and see if it satisfies it.

I hope this explanation has helped clarify the concept of extraneous solutions in