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# Solving absolute value inequalities: no solution

Sal solves the inequality |y|+22 ≤ 13.5 to find that it has no solution. Created by Sal Khan and Monterey Institute for Technology and Education.

Video 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.