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Absolute Value Equations - Problem 2
The first thing to do when solving an absolute value equation is to isolate the absolute value quantity. The rules that apply to isolating the variable in an equation apply to isolating the absolute value quantity. In other words, eliminate everything else on the same side of the equal sign except for the absolute value quantity using inverse operations. Be mindful of keeping track of all the signs. Next, split the equation into two equations and remove the absolute value signs. One equation should be equal to the positive quantity on the other side, and the second equation should be equal to the negative quantity. Solve for the variable in both equations. Don't forget to check your solutions by substituting them back into the original equation. Remember that if the absolute value quantity is equal to a negative number, then there is no solution to the equation since an absolute value cannot equal a negative number.
When you're solving absolute value equations, again it's important to think in mind how many solutions you might have. Usually you are going to have two and it's also important to be really careful showing your work with the negative signs because it's easier to get messed up.
Like in this problem for example I have this -2 times the absolute value quantity. I'm going to have to deal with that when I divide both sides by -2.
First though I'm going to subtract 3 from both sides so that I have -2 times the absolute value of x take away 3 is equal to -8. Here's where that negative sign becomes tricky. I'm going to divide both sides by -2 keeping in mind my goal is to get the absolute value piece all by itself so that I'll have now absolute value of x take away 3 is equal to negative divide by negative is positive 4.
At this point I can check and see am I going to have two solutions, one solution or zero solutions? And the way you can tell is by looking to see if your absolute value quantity is equal to a regular number like positive if it's equal to 0 you're only going to have one answer or if it's equal to a negative value here you are going to have no solutions.
In our case I can tell we are going to have two solutions because my absolute value is equal to +4. In order to find both of those solutions I'm going to have to solve x minus 3 equals 4 and also x minus 3 equals -4. This minus sign that was inside the absolute value stays as a minus sign but this 4 in one situation stays as positive, in the other situation it becomes negative.
Go through and solve for x, you get x equals 7 and also x equals -1. A lot of students get kind of confused because they say, "Wait a second you said absolute value couldn't be negative you can't have a negative distance. That's true this absolute value piece would be, have no solutions if it was equal to a negative value. However x can be negative when I put -1 in there I'm going to absolute value-ize it and it's going to end up becoming a positive and that's okay.
So again the most important thing to keep in mind with absolute value is you have the keep track of your negative signs and you also probably want to stop and think about how many solutions you'll have when you get to the point where your absolute value piece is isolated.