# Absolute value inequalities | Linear equations | Algebra I | Khan Academy

I now want to solve some
inequalities that also have absolute values in them. And if there’s any topic in
algebra that probably confuses people the most, it’s this. But if we kind of keep our head
on straight about what absolute value really means, I
think you will find that it’s not that bad. So let’s start with a nice,
fairly simple warm-up problem. Let’s start with the absolute
value of x is less than 12. So remember what I told
you about the meaning of absolute value. It means how far away
you are from 0. So one way to say this is, what
are all of the x’s that are less than 12 away from 0? Let’s draw a number line. So if we have 0 here, and we
want all the numbers that are less than 12 away from 0, well,
you could go all the way to positive 12, and you could go
all the way to negative 12. Anything that’s in between these
two numbers is going to have an absolute value
of less than 12. It’s going to be less
than 12 away from 0. So this, you could say, this
could be all of the numbers where x is greater
than negative 12. Those are definitely going to
have an absolute value less than 12, as long as they’re
also– and, x has to be less than 12. So if an x meets both of these
constraints, its absolute value is definitely going
to be less than 12. You know, you take the absolute
value of negative 6, that’s only 6 away from 0. The absolute value of negative
11, only 11 away from 0. So something that meets both
of these constraints will satisfy the equation. And actually, we’ve solved it,
because this is only a one-step equation there. But I think it lays a good
foundation for the next few problems. And I could actually
write it like this. In interval notation, it would
be everything between negative 12 and positive 12, and not
including those numbers. Or we could write it like this,
x is less than 12, and is greater than negative 12. That’s the solution
set right there. Now let’s do one that’s a little
bit more complicated, that allows us to think
a little bit harder. So let’s say we have the
absolute value of 7x is greater than or equal to 21. So let’s not even think about
what’s inside of the absolute value sign right now. In order for the absolute
value of anything to be greater than or equal to
21, what does it mean? It means that whatever’s inside
of this absolute value sign, whatever that is inside of
our absolute value sign, it must be 21 or more
away from 0. Let’s draw our number line. And you really should visualize
a number line when you do this, and you’ll never
get confused then. You shouldn’t be memorizing
any rules. So let’s draw 0 here. Let’s do positive 21, and let’s
do a negative 21 here. So we want all of the numbers,
so whatever this thing is, that are greater than
or equal to 21. They’re more than
21 away from 0. Their absolute value
is more than 21. Well, all of these negative
numbers that are less than negative 21, when you take their
absolute value, when you get rid of the negative sign,
or when you find their distance from 0, they’re all
going to be greater than 21. If you take the absolute value
of negative 30, it’s going to be greater than 21. Likewise, up here, anything
greater than positive 21 will also have an absolute value
greater than 21. So what we could say is 7x needs
to be equal to one of these numbers, or 7x needs to
be equal to one of these numbers out here. So we could write 7x needs to
be one of these numbers. Well, what are these numbers? These are all of the numbers
that are less than or equal to negative 21, or 7x– let me do a
different color here– or 7x has to be one of
these numbers. And that means that 7x has to
be greater than or equal to positive 21. I really want you to
kind of internalize what’s going on here. If our absolute value is greater
than or equal to 21, that means that what’s inside
the absolute value has to be either just straight up greater
than the positive 21, or less than negative 21. Because if it’s less than
negative 21, when you take its absolute value, it’s going to
be more than 21 away from 0. Hopefully that make sense. We’ll do several of these
practice problems, so it really gets ingrained
in your brain. But once you have this set up,
and this just becomes a compound inequality, divide both
sides of this equation by 7, you get x is less than
or equal to negative 3. Or you divide both sides of
this by 7, you get x is greater than or equal to 3. So I want to be very clear. This, what I drew here, was
not the solution set. This is what 7x had
to be equal to. I just wanted you to visualize
what it means to have the absolute value be greater
than 21, to be more than 21 away from 0. This is the solution set. x
has to be greater than or equal to 3, or less than
or equal to negative 3. So the actual solution set to
this equation– let me draw a number line– let’s say that’s
0, that’s 3, that is negative 3. x has to be either greater
than or equal to 3. That’s the equal sign. Or less than or equal
to negative 3. And we’re done. Let’s do a couple
more of these. Because they are, I think,
confusing, but if you really start to get the gist of what
absolute value is saying, they become, I think, intuitive. So let’s say that we have
the absolute value– let me get a good one. Let’s say the absolute value of
5x plus 3 is less than 7. So that’s telling us that
whatever’s inside of our absolute value sign has to be
less than 7 away from 0. So the ways that we can be less
than 7 away from 0– let me draw a number line– so the
ways that you can be less than 7 away from 0, you could be less
than 7, and greater than negative 7. Right? You have to be in this range. So in order to satisfy this
thing in this absolute value sign, it has to be– so the
thing in the absolute value sign, which is 5x plus 3–
it has to be greater than negative 7 and it has to be less
than 7, in order for its absolute value to
be less than 7. If this thing, this 5x plus 3,
evaluates anywhere over here, its absolute value, its
distance from 0, will be less than 7. And then we can just
solve these. You subtract 3 from
both sides. 5x is greater than
negative 10. Divide both sides by 5. x is
greater than negative 2. Now over here, subtract
3 from both sides. 5x is less than 4. Divide both sides by 5, you
get x is less than 4/5. And then we can draw
the solution set. We have to be greater than
negative 2, not greater than or equal to, and
less than 4/5. So this might look like a
coordinate, but this is also interval notation, if we’re
saying all of the x’s between negative 2 and 4/5. Or you could write it all of the
x’s that are greater than negative 2 and less than 4/5. These are the x’s that satisfy
this equation. And I really want you
to internalize this visualization here. Now, you might already be seeing
a bit of a rule here. And I don’t want you to just
memorize it, but I’ll give it to you just in case
you want it. If you have something like f of
x, the absolute value of f of x is less than, let’s
say, some number a. Right? So this was the situation. We have some f of
x less than a. That means that the absolute
value of f of x, or f of x has to be less than a away from 0. So that means that f of x has to
be less than positive a or greater than negative a. That translates to that, which
translates to f of x greater than negative a and f
of x less than a. But it comes from
the same logic. This has to evaluate to
something that is less than a away from 0. Now, if we go to the other side,
if you have something of the form f of x is
greater than a. That means that this thing has
to evaluate to something that is further than a away from 0. So that means that f of x is
either just straight up greater than positive a, or f of
x is less than negative a. Right? If it’s less than negative a,
maybe it’s negative a minus another 1, or negative
5 plus negative a. Then, when you take its
absolute value, it’ll become a plus 5. So its absolute value is going
to be greater than a. So I just want to– you could
memorize this if you want, but I really want you to think about
this is just saying, OK, this has to evaluate, be less
than a away from 0, this has to be more than a away from 0. Let’s do one more, because
I know this can be a little bit confusing. And I encourage you to watch
this video over and over and over again, if it helps. Let’s say we have the absolute
value of 2x– let me do another one over here. Let’s do a harder one. Let’s say the absolute value
of 2x over 7 plus 9 is greater than 5/7. So this thing has to evaluate to
something that’s more than 5/7 away from 0. So this thing, 2x over 7 plus
9, it could just be straight up greater than 5/7. Or it could be less than
negative 5/7, because if it’s less than negative 5/7, its
absolute value is going to be greater than 5/7. Or 2x over 7 plus 9 will be
less than negative 5/7. We’re doing this case
right here. And then we just solve both
of these equations. See if we subtract– let’s just
multiply everything by 7, just to get these denominators
out of the way. So if you multiply both sides by
7, you get 2x plus 9 times 7 is 63, is greater than 5. Let’s do it over here, too. You’ll get 2x plus 63 is
less than negative 5. Let’s subtract 63 from both
sides of this equation, and you get 2x– let’s see. 5 minus 63 is 58, 2x
is greater than 58. If you subtract 63 from both
sides of this equation, you get 2x is less than
negative 68. Oh, I just realized I
made a mistake here. You subtract 63 from both sides
of this, 5 minus 63 is negative 58. I don’t want to make a careless
mistake there. And then divide both
sides by 2. You get, in this case, x is
greater than– you don’t have to swap the inequality, because
we’re dividing by a positive number– negative 58
over 2 is negative 29, or, here, if you divide both sides
by 2, or, x is less than negative 34. 68 divided by 2 is 34. And so, on the number line,
the solution set to that equation will look like this. That’s my number line. I have negative 29. I have negative 34. So the solution is, I can either
be greater than 29, not greater than or equal to, so
greater than 29, that is that right there, or I could be
less than negative 34. So any of those are going to
satisfy this absolute value inequality.

## 100 thoughts on “Absolute value inequalities | Linear equations | Algebra I | Khan Academy”

1. Salma Larhlimi says:

jazaka allah khayran

2. Eyoel Agaze says:

get your Closed Captioning but out of here

3. Elijahjavierr03 says:

When you don’t pay attention in class

4. Awesome Guy says:

Thank you so much! I understand now:)

5. Instrument Hoarder says:

how do you know if its and or or

6. BlitzChris says:

You people are lifesavers

7. Angry prash says:

hello sir I I'm shashi and see your videos which is very important for me sir now I have to faced problem in inequalities of modules with fraction so please help me and to making so useful video

8. Arnav Verma says:

#YouCanLearnAnything

9. Elham says:

Wow u should be a teacher

10. Faisal Maniar says:

explicit. Thanks sal.

11. Caleb Johnson says:

you saved me from my algebra class thankyou

12. Little Jimmy Fitness Advice says:

math exam tomorrow, and u taught this so well that after the first number line i could do the most complicated absolute values. Thanks man

13. Yung Trixx says:

Horrible

14. Amir says:

Than youuuuuu
You were the only person that brought tears of joy on my cheeks in the middle of tears of unhappiness (FINALS!)

15. D C says:

Still wondering about All Real No.s and no solution tho? Can someone teach me

16. Ritvik Singh says:

this is very helpful

17. Edgar Lopez says:

thanks

18. Antweezy says:

what about union and intersection?

19. clarizza f says:

i love khan academy sm

20. Lumine_Kim says:

HEEEEEY

21. Lumine_Kim says:

MEKENI MEKENI DUGDUG DOREMI !!!

22. Lumine_Kim says:

i love khan academy so much but

you're road, you know?

23. clarizza f says:

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24. bloop bloop says:

My problem in math is nothing like this it's. ➡️➡️ |x-2| < 10. Does anyone know how to do this 🤔

25. Mritunjay says:

First 9th grade then 10th and now current 11th.. text books changed ..teachers changed..school changed ..but sal is still there ..thanks a lot..

26. Ajapa Saha says:

Who else has the khan academy up but is still watching it on youtube

27. Ujan Timsina says:

Khan academy is waste of time!

28. curious learner says:

Very well explained, guys if you don't get this first study the concept of graphical solutions of inequality and intervals then u will surely understand this

29. ayush patil says:

Thank you. This is helpful

30. Shuvhashish Saha says:

Thank you so much for this video 🙂

31. Rituraj Mahto says:

Awesome teaching

32. Booga04 Minecraft says:

You're so helpful. Thanks, Khan!

33. Sean Mcneme says:

Omg I've gone my whole life without understanding absolute value correctly. I'm studying for the GRE and was really struggling with these problems. Great video!

34. puppet master says:

man…. wich set works? dose both of them work? can somedody anser this repliy to me pleaze

35. Kaitlyn Espinoza says:

this was literally no help at all.

36. David Jan says:

thank god for sal

37. Mr. [EXPUNGED] says:

What do I do if the problem is something like: |x-3| = 1

38. Serpens says:

thank you SO MUCH!! this was extremely helpful

39. Catherine Brooke says:

as a 7th grader taking math 1 i can confidently say…
i’m gonna die on my quiz tom… yay

40. Gregory Staniszewski says:

if there is an x value on the right side of the greater to or equal than sign, do i carry it over first before I consider the 2 or options and make sure Xs are on one side and it's greater to equal to 0?

41. Pickled Beef says:

This is actually better than my math teacher

42. sandeep rao says:

Plz tell why did u write lxl=12 as x less than -12

43. Joud AlGhamdi says:

dude thanks

44. Savithma Dinandi says:

Thank you a lot ,I could understand everything as the way you teach is excellent.

45. True Tekker says:

This is why Khan Academy is the boss!

46. Ali Gams says:

thanks

47. Batool Alshuala says:

Thank you 😭♥️🙏🏻

48. Nicholas Ouellette says:

i put 10 of this same video in different tabs and its great. THANKS KHAN ACADEMY!!!!!

49. Kendrick Jude Mausisa says:

Thanks for the clarification Sal! Now I understand why the intervals point where!

50. ismael palomino says:

In class we went over this for two weeks and never understood it until I watched this video and finally got the hang of it

51. jis morgan says:

CONFUSING

52. Ricardo Garcia says:

So then what is the greatest value of absolute value x+3 less than or equal to 2…
1. -1
2. -2
3. -3
4. -4
5. -5

53. GAMING says:

Wow nice video

54. Afrin Atiq says:

#Thanks

55. Maria de Lourdes Schneider says:

You're an awesome educator! thank you!!

56. •Mirazkie Vlogs• says:

im only grade 7 but this is our lesson right now.

57. Strawberry098 says:

Thank you so much, your explanation was easy to understand. Thxxxx!

58. John Doe says:

wow. this is so easy after watching the first 1 minute. thanks!

59. Karem Hashem says:

thanks partner

60. Erik Rodriguez says:

what if there is no solution how does the graph look?

61. The Anonymous says:

Oh yeah yeah

62. Angelo Palma says:

Who is actualy a 1st grader?

63. Jimmy Delano says:

Okay, but like the first problem: X is less than 12. You write the solution: (-12,12) Yes, -12 is less than 12. But -13 is also less than 12. -14 is also less than 12. -15 is also less than 12, ect. So technically, shouldn't the the solution be [negative infinity, 12)? I just don't understand why you choose negative 12 and stop there. This is not the way the way I was taught.

64. Ruffy says:

i can’t understand my teachers teaching but this video actually made me get the hang of it

65. Pamela Elliott says:

One primary complaint is incorrect wording – using the word "equation" when should be saying "inequality". This verbiage error occurs multiple times throughout the video.

66. Hi Lo says:

What I do in my free time

67. Luh. Alianaaa says:

Yall pray for me I have a science test (forces) and a algebra test (absolute values) tommorow !!

68. Audrey POWERS says:

lol while is everyone studying for finals, I'm just tryna do ok in 7th-grade maths

69. Ritesh Gamer says:

wow I understood it now. no memorizing now

70. Noah Harris says:

Thumbs up if your summer semester Calc II

71. Padmavathy Krishnamoorthy says:

Why can't you be my math teacher?? I have a math teacher who dont know what i know in math. Education is not equally distributed among the world.

72. Sonia Martinez says:

Doing and AP stats review packet and I don’t get the first question so now I’m here 🤪

73. mahfooz alam says:

Thank you sir

74. shruti jain says:

Thanq so much!!

75. star gamer says:

your way better then my teacher

76. MODERN Gamer says:

Anyone from 9th

77. Yasmiena says:

this was super helpful im in algebra 2 and we are learning this and it made no sense with the teacher but now it makes so much more sense thank you

78. Infamous Pineapple says:

This didn’t help

79. Aarushi Agarwal says:

Omg I wish all school teachers taught like this. Sal makes everything easier. I love him.

80. lol Clan says:

and he writes this all with a mouse

I think

81. Maryam says:

god bless you.

82. NïghtMäreBluue says:

how did he get the 5???!!!!

83. I’m a failure says:

Khan i love you

84. Emmanuel Moreno says:

How do you graph 7|5p-7|=-21

85. Nick 247 says:

I think this is great. I had all the rules and stuff but it really helps you when you understand what is happening. Im so much better at this.

86. Isaac Joseph says:

10 years later still saving lives

87. Khaled B says:

My teacher doesn’t teach me. Sal does.

88. Victor Rodriguez says:

This couldn't have being explained any clearer.

89. christianne delos santos says:

oof I have a test in 10 minutes and I didn’t study

90. Forklift17 says:

8:56 should be "and", not "or"

91. lunacus5 says:

I’m doing this in the beginning of eight grade all you high school nerds

92. Hitless KD says:

Its ironic that if school teachers were good and explained the topic good this channel wouldn't be as popular or even exist.

93. OutdoorFilmsTV ! says:

You didn’t put the open dot on 7 and -7

94. Faisal-_- A7MD says:

Tomorrow I got an exam on this lesson

Wish me luck👍🙂

95. P M says:

🌸🌸

96. Shivam Goswami says:

this is so nicely explained. thank you sir

97. Justice For Maximilianmus says:

I'm ready for my test now

98. Mahmoud Gaming says:

i didn't understand the 11 minute

99. LJM Coppell says:

Can anyone help me figure out what to do if this is the problem

|-2c-3|>-4

100. nas tay says:

bro I'm already a sophomore and I still didn't know how to solve absolute value inequalities. thank you so much this really helped a lot. I literally forgot everything I learned in middle school and my freshman year