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Absolute Value Equations - Problem 1
To solve an absolute value equation, start by isolating the absolute value. Just like you would isolate a variable using inverse operations, the same rules apply to isolate the absolute value -- eliminate everything else on the same side of the equal sign except for the quantity inside the absolute value signs using inverse operations. 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.
This is the kind of problem where you're asked to solve for x in an absolute value. The first thing you want to do is isolate this absolute value piece before you do any solving. Also keep in mind you’re probably going to have two answers so when it ask you to check your solution it should probably actually say check your solutions, because you’re probably going to have two answers.
So let's try. Okay so to isolate the absolute value first thing I need to do is divide by 3 so that that 3 piece is gone away so it will look like this. Absolute value of x take away 4 is equal to 4.
Now that my absolute value is isolated I’m ready to split them I’m going to actually have two different problems. I need to solve x take away 4 equals to 4 and x take away 4 equals -4. That's going to help me get my two answers because remember with absolute value it means distance away from zero. We have to account for both the positive direction and the negative direction.
Go through and you got to do two problems now, you’ve got to solve for x where is equal to +4 you also have to solve for x where it's equal to -4, oops plus 4, plus 4 x is equal to 0. Those are my two answers I think I move on though I want to point out something that I did, that was kind of tricky.
When you have this absolute value piece and you're splitting it to be equal to +4 and -4, notice that the positive and negative change on this 4 here like I have a negative one there, positive one there. This x minus 4 stays x minus 4 for both problems. I'm not messing with whatever was inside the absolute value piece, I'm only changing the sign of what was outside.
Okay so let’s go ahead and check the solutions. The way to check is to go back to the original problem and substitute in each value separately. So here is my I check for when x is equal to 8 let's see if it’s true that 3 times the absolute value of 8 take away 4 is equal to 12. Then you’ll see that in fact yes, 3 times the absolute value 4 is equal to 12 and then let's also check when x is equal to 0. I'll have 3 times 0 take away 4 inside the absolute value I hope that's equal to 12 and you'll see it isn't that because the absolute value of negative 4 is positive 4, 3 times 4 equals 12.
This is a situation where you have two solutions it’s really important to check your work to make sure you did it correctly and didn't make any funny mistakes with the plus or minuses. When you have problems like this you have to be really careful working through your positive and negative answers.