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## Algebra (all content)

### Course: Algebra (all content)ย >ย Unit 8

Lesson 3: Solving absolute value inequalities- Intro to absolute value inequalities
- Solving absolute value inequalities 1
- Solving absolute value inequalities 2
- Solving absolute value inequalities: fractions
- Solving absolute value inequalities: no solution
- Absolute value inequalities word problem

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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.

## Want to join the conversation?

- Can you put a negative sign outside the absolute value sign?(18 votes)
- Good question! Yes you can. The graph f(x) = -lxl looks exactly like f(x) = lxl except it is flipped on the x-axis. In other words everything will be negative (or zero) no matter what number is inside.

I hope this helps!(23 votes)

- When Sal says there is no solution, he means, there is no solution in real numbers? its possible exists a solution in complex numbers to solve the inequality?(8 votes)
- No. Even when we extend the notion of absolute values to the complex numbers, the values never become negative. Just like with the imaginary numbers themselves, you'd have to artificially define a new number with an absolute value of -1 if you thought that finding formal solutions of equations like this were worth having.(14 votes)

- What if the inequality has a negative absolute value, but the variable is greater than? Such as k>-5 Is this still no solution? Because there are positive numbers that are above -5, they're just not -5.(6 votes)
- Yes,
`| k | > -5`

has a solution set, but`| k | < -5`

does not since the absolute value cannot be negative.(11 votes)

- how can I solve double absolute value equations?(5 votes)
- Hi,

I would just like to know if there are any practice questions associated with this video. If so, please give me a link. Thanks!(3 votes)- There aren't any specifically associated practice questions with this video in particular, but there is general absolute value equations that you can practice, here's the link: https://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/absolute-value-equations/e/absolute_value_equations

Have fun! :}(4 votes)

- I have a double absolute value inequation: |2x| < |x-5|. I don't know how to proceed. Is there a video about something like this problem?(2 votes)
- As far as I can tell, Khan doesn't have a video dealing with this specific avenue yet =/ This site goes over it very briefly

http://brightstorm.com/math/precalculus/linear-equations-and-inequalities/solving-absolute-value-equations

Under problem 3 though.

To solve your inequality, you can try to form equations where both abs values are positive, and then when one is positive and the other is negative

So

1 (Both are positive). Sign changes because 2x is always > x-5 when x is positive

2x > x-5

x > -5

2.(One is positive, other is negative)

-2x < x-5

-3x < -5

x > 5/3

So -5 < x < 5/3

Honestly, it would probably be best to graph X)(4 votes)

- I can not figure out how to solve an inequality with two numbers outside the absolute value sign. For instance, how would I solve 4|2w+3| - 7< 9?(2 votes)
- First, you have to isolate the absolute value on one side. Add 7 to both sides and divide both sides by 4 to get |2w+3|<4. Then, you can solve it just like in the video.(2 votes)

- Can you make a video explaining how to solve inequalities with the two absolute values on the same side?

Such as: | x+1 | + | x+1 | less than or equal to 2(2 votes) - I am also having trouble solving an inequality with two absolute value functions. I am not aware of any videos about this topic on Khan Academy therefore I am enquiring as to whether anyone has any websites that explain the methods for solving such inequalities. the website rynkwn suggested has an internal error. the questions i am trying to solve is: find the set of values of x for which | x โ 1 | > | 2x โ 1 |. one suggested method is squaring both sides...(1 vote)
- As you may have seen from other replies, for solving such problems you have to divide the equation into "regimes", based on the expression(s) of x that are enclosed in absolute value brackets.

Based on your equation, we have three regimes:

(i) x >= 1 (ii) 1/2 <= x < 1 (iii) x < 1/2

For (i): the equation becomes x - 1 > 2x - 1, giving x < 0

However the assumption was x >= 1. Since the assumption is inconsistent with the solution, x>=1 is not a solution.

(ii) For this regime, the equation becomes: 1-x > 2x - 1

This gives the solution x < 2/3; If we combine with our regime assumption, we get the solution set as 1/2 <= x < 2/3

(iii) For this regime, our equation is: 1 - x > 1 - 2x

This gives the solution set x > 0. If we combine this with our regime assumption, we get the solution set as: 0 < x < 1/2

THerefore the final solution set, combining results of (ii) and (iii) is:

0 < x < 2/3. You can take random values within this regime to make sure the solution set satisfies the inequality.(3 votes)

- What does Sal mean when he says:[a]= x is a non negative number or zero ??(2 votes)
- If the absolute value of a = x, then this means x can be a positive number or zero. This is because the absolute value of something can never be a negative number.(2 votes)

## 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.