Solving absolute value inequalities: no solution
About this video
Sal solves |y| + 22 <= 13.5 by subtracting 22 to get |y| <= -8.5. Since the absolute value of any number is always 0 or positive, it can never be less than or equal to a negative number, so the inequality has no solution.
You'll be able to
- isolate an absolute value in an inequality.
- explain why an absolute value is always non-negative.
- recognise when an absolute value inequality has no solution.
Key words
- non-negative
- Either 0 or positive; never below 0.
- no solution
- No value of the variable makes the statement true.
- absolute value
- A number's distance from 0.
Transcript
We're told to solve for y, and we have this inequality that says that the absolute value of y plus 22 is less than or equal to 13 and 1/2 or 13.5. So a good place to start is maybe to just isolate the absolute value of y on the left-hand side of this inequality.
And the best way to do that, we can subtract 22 from both sides. So let's subtract 22 from both sides. The left-hand side, these guys cancel out, that was the whole point, so you're just left with the absolute value of y as being less than or equal to. And then 13.5 minus 22, let me do it over here, 13.5 minus 22.
My brain imagines, or the way I process it is, I say, well-- I always like to put the larger number first-- I say that's the negative of 22 minus 13.5. And 22 minus 14 is 8, or the difference between 22 and 14 is 8, so the difference between 22 and 13 and 1/2 is going to be 1/2 more than that.
So this is going to be 8.5. So it's going to be negative 8.5. So we get the absolute value of y is less than or equal to negative 8.5. Now, this should cause you some pause, because when you take the absolute value of anything, what do you know you're going to get?
If I tell you that the absolute value of any number, oh, we'll just say the absolute value of a is equal to x. What do you know about x? You know that x is non-negative. It's either a positive number or 0. Here we're saying that y is, when we take the absolute value, has to be less than or equal to a negative number.
It's saying that it has to be a negative number. This is implicitly saying that the absolute value of y has to be negative. It not only has to be negative, it has to be less than or equal to negative 8.5. We know that if you take the absolute value of anything, you're going to get 0 or a positive number.
You're going to get a non-negative number. There's no number you can put here whose absolute value's going to give you a negative number, especially one less than negative 8.5. So there is no solution to this problem. You cannot find a y that will satisfy this.
So there is no solution.
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