Why Some Solutions to Absolute Value Equations are FalseDate: 10/24/2004 at 19:09:23 From: Dan Subject: Why are some values rejected when solve absolute value eq's? Hi there, I'm curious as to why some values are rejected when solving absolute value equations. I understand that this only occurs when you have checked your solution and one of the values turns out to be false, but my question is WHY the equation can yield false values. An example of this would be the following question: |4x+3| + 21 = 5x which yields the possible x-values of 24 and 2. When you check those values, 2 turns out to not work. WHY? Any help as to why this can happen would be greatly appreciated. Date: 10/24/2004 at 21:30:04 From: Doctor Rick Subject: Re: Why are some values rejected when solve absolute value eq's? Hi, Dan. Let's trace through the solution and then trace through what happens when we plug in the spurious solution. First, my solution: |4x+3| + 21 = 5x |4x+3| = 5x-21 4x+3 = 5x-21 OR -(4x+3) = 5x-21 21+3 = 5x-4x -4x-3 = 5x-21 24 = x -3+21 = 5x+4x 18 = 9x 2 = x So we have two (tentative) solutions: x = 24 or 2. Now we'll plug each solution into the original equation and evaluate: |4x+3| + 21 = 5x |4x+3| + 21 = 5x |4*24+3| + 21 = 5*24 |4*2+3| + 21 = 5*2 |99| + 21 = 120 |11| + 21 = 10 99 + 21 = 120 11 + 21 = 10 120 = 120 32 = 10 CORRECT WRONG Why did the second solution fail? It would have worked if the absolute value of 11 were -11; then the third line would read -11 + 21 = 10 which is correct. The problem arose because the absolute value of a quantity is the negative of that quantity ONLY IF the quantity is negative. In other words, a more formal definition of absolute value is: { x if x > 0 |x| = { 0 if x = 0 {-x if x < 0 This was not taken into account in my solution method. In the third line of the solution, I said that the absolute value could be either the quantity itself, 4x+3, or its negative, -(4x+3). I didn't check at that stage whether the quantity (4x+3) was negative or positive. I hold off this check until after I have found the candidate solutions. This is why the testing of the candidate solutions is an integral part of the solution process, and not just a double-check on my work. Does this help? - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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