College Algebra: Absolute Value Inequality Example

This lesson has not been reviewed.

Please purchase the lesson to review.

Let's take a look at some more absolute value inequalities. I think these things are actually worth a couple of extra practice problems just to get you sort of in the swing of things. This one is going to look a little different and you subtract off 4 and that has to be less than 10. So, that's the question. Now, the issue is how do you deal with that. Well, the first thing that I do is just get the absolute value by itself. It's like we see often a lot as a theme. Let's get the thing that we're concerned about or that's sort of a unit by itself. So, I'm going to bring this 4 over and make it therefore 14. I have < 14. Then I think to myself now what does it mean for an absolute value to be less than something. It means that it's going to be small in size.
So, if it's going to be small that means I'm thinking like this. I'm just going to draw a little sketch of what I'm thinking. It means that it's going to be within 14 or -14. So, if I put in 14, this is not drawn to scale at all, then I know that I'm living somewhere in here. See, just by thinking like this, it helps me set up the right inequality, because I realize therefore I'm going to want this thing to be a bigger than -14, but also less than 14. This little picture actually helps a lot because it allows me to immediately say that is the exact same thing as -14 < 5y + 2 <14. You see, this is my thinking always. I think about what it means for an absolute value to be small or if the question was absolute value big, what that means. I sketch a little teeny picture to help me and then I set it up.
Now, I'm going to solve this inequality. What do I do? Well, I'm going to subtract two everywhere. If I subtract 2 from here, I have -16, same inequality. 5y, if I subtract 2 from here, that's gone, same inequality. If I subtract 2 from here I have 12. Now, I divide everything through by five. No need to flip all those inequality signs because five is positive. When I do that I see that I have -16 over 5 is less than y, which in turn is less than 12 over 5 and that's the answer. What does that mean? It means that any single y that's bigger than -16 over 5, yet at the same time less than 12 over 5, any such y will satisfy this original inequality. Any y outside of that little region, any y outside of that, I know will not satisfy this. So, this, the solution to this absolute value inequality.
How about one more? In fact, this one, I'll five you a shot at this one. These things take a little bit or time, but practicing them is really a worthwhile thing. So, let me really encourage you to try to think about this. See if you can solve this one and I'll be back and we'll see if I can solve it. Give it a shot.
Well, this is sort of a trick problem in a way. Let me talk through it first and then let's figure out the answer using a sneaky idea and then I'll work through it as we would have normally. Look, we know that the absolute value of anything is always positive or zero. So, in fact, I know that if I put anything in here it's always going to be greater than or equal to zero. Here I'm asking that this thing has to be greater than zero. What does that mean? Well, that means that this inside thing can be any number at all as long as it's not zero itself. So, what would make this thing zero? Well, x = 3. So, as long as x does not equal three I know I'm home free. So, the answer must be all numbers but three.
So, that was sneaky because this thing is positive. Now, how would you have done that if you wouldn't have seen that sneaky thing? Well, then you would just think as I think. Here's how I would have thought about it if I didn't notice that sneaky little thing. I'd say "Well, this absolute value is sort of repelled away from zero, so it's going to be bigger than zero but also less than minus zero." Well, zero and minus zero are the same. So, it would look like this. It'd be really weird looking. Here's zero. It would have to live way out here or way out here. Do you see it's exactly what we said before? It would be every single possible number except zero.
So, now I would actually have to technically solve this inequality, which would be--I'll write it out here, 2x - 6 > 0 and this inequality, which would be 2 - 6 < 0. If you solve this inequality, which I'll let you try on your own, well, you've already tried it I hope, you'll see that x < 3. The solution to this inequality is x > 3. So, then what's the answer? Well, either x < 3 or x > 3. What does that mean? It means that x can be any single number at all except three. It's either bigger than three or less than three, but not equal to three.
A little trick problem there, but I thought it'd be fun. I thought, "Gee whiz, throw a little joke and humor." Happily that's the last inequality problem I'm going to do now. So, we'll take a little break here and we'll come back and do something else.
Equations and Inequalities
Inequalities- Absolute Values
Solving Absolute Value Inequalities: More Examples Page [1 of 1]