The rule for rewriting an absolute value equation without absolut...

Question

The rule for rewriting an absolute value equation without absolute value bars can be extended to equations with two sets of absolute value bars:
If u and v represent algebraic expressions, then |u| = |v| is equivalent to u = v or u = - v. Use this to solve the equations in Exercises 77–84.

|3x - 1| = |x + 5|

Accepted Answer

Hello. Everyone in this video we're going to be looking at this practice problem where we want to solve the equation, absolute value of two X minus 10 is equal to the absolute value of X plus one. So in this case we're going to be working with absolute value. So it's important to remember that when you have the absolute value of a function you're always going to get a positive number. So say I was taking the absolute value of you I would always get a positive A. However you could really be positive A. Or it could be negative A. Because remember that the absolute value only takes the magnitude of the number inside. So I'm going to take that into account when I'm solving for this absolute value equation By saying two X -10 could be equal to x plus one or it could be equal to two X minus 10. Could be equal to negative X plus one. So that's taking into account if two X -10 is negative or positive. So I'm gonna solve if it was positive first I'm gonna get X on one side. So I'm going to add 10 and subtract X. So it leaves me X is equal to 11. So that simplifies if the value inside the absolute value is positive. Now I'm going to do the same for if the value was negative Where I have two X -10 is equal to distribute the negative negative x minus one. So I'm gonna add X to both sides and add 10 to both sides. That leaves me with three X is equal to nine. If I divide by three I have X is equal to three. So now I have two values for X. X equals 11 and X equals three. So in order to make sure that these are valid solutions, I'm gonna plug them back into my equation two X minus 10 is equal to X plus one. So starting with x equals 11. I plug that in. I have absolute value of two times 11 -10 Is equal to the absolute value of 11-plus 1. So I have the absolute value of 22 -10 Is equal to the absolute value of 12. So 22 -10 is equal to and the absolute value of 12 is equal to 12. So I'm left with 12 is equal to 12 and that is true. So I know that X equals 10 or X equals 11 is a valid solution. Now I'm going to do the same thing, x equals three. So if I plug in three for my ex I have the absolute value of two times three -10 is equal to the absolute value of 3-plus 1. So now I have 6 -10 is equal to four and 6 -10 is equal to negative four and the absolute value of four is four and then the absolute value of negative four is four. So now I have four is equal to four and that is also true. So I know that X equals three is another Solution that works. So now that I've verified that both of them work, I know that X could be 11, or X could be three. So that becomes my final answer. And you can see that that aligns with answer choice A. So hopefully this video helped and see you next time.

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