# Algebra: Solving Inequalities

Welcome to the presentation
on solving inequalities, or I guess you call them
algebra inequalities. So let’s get started. If I were to tell you that,
well, let’s just say x is greater than 5, right? So x could be 5.01, it could be
5.5, it could be a million. It just can’t be 4, or 3, or 0,
or negative 8, and actually, just for convenience,
let’s actually draw that on the number line. That’s the number line. And if this is 5, x can’t be
equal to 5, so we draw a big circle here, and then we would
color in all the values that x could be. So x could be just the small–
it could be 5.0000001. It just has to be a little bit
bigger than 5, and any of those would satisfy it, right? So let’s just write some
numbers that satisfy. 6 would satisfy it, 10
would satisfy it, 100 would satisfy it. Now, if I were to multiply, or
I guess divide, both sides of this, I guess we could say,
equation or this inequality by negative 1, I want to
understand what happens. So what’s the relation between
negative x and negative 5? When I say what’s the relation,
is it greater than or is it less than negative 5? Well, 6 is a value
that works for x. So negative 6, is that greater
than or less than negative 5? Well, negative 6 is less
than negative 5, right? So let me draw the
number line here. If we have negative 5 here, and
let’s just draw a circle around it because we know it’s not
going to be equal to negative 5 because we’re deciding between
greater than or less than. So we’re saying 6 works for x. So negative 6 is here, right? So negative 6 is less than
negative 5, so is negative 10, so is negative 100, so is
negative a million, right? So it turns out that negative
x is less than negative 5. And this is really all you have
to remember when you are working with inequalities
in algebra. Inequalities, you can treat
them just the way– a greater than or less than sign, you
could treat them exactly the way you would treat
an equal sign. The only difference is, if you
multiply or divide both sides of the equation by a negative
number, you swap it. That’s all you
have to remember. Let’s do some problems,
and hopefully, that’ll hit the point home. And if you ever forget, you
just have to try– you just remember this: if x is greater
than 5, well, then negative x is less than negative 5. And keep trying out numbers. That’s what’s going to give
you the best intuition. Let’s do some problems. So if I said that 3x plus 2 is,
let’s say, less than or equal to 1– well, this is a pretty
easy equation to solve. We just say 3x– let’s subtract
2 from both sides, and when you add or subtract, you don’t do
anything to the inequality. So if you subtract 2 from both
sides, you get 3x is less than or equal to negative 1, right? And then, now we’re going
to divide both sides by 3. We get x is less than or equal
to negative 1/3, right? And notice, we didn’t
change anything because we divided both sides
by a positive 3, right? We could have actually
done this equation in a slightly different way. What if we subtracted
1 from both sides? So this is another
way of solving it. What if we said 3x plus 1 is
less than or equal to 0, right? I just subtracted 1 from both
sides, and now let’s subtract 3x from both sides. And we get 1 is less than or
equal to minus 3x, right? I subtracted 3x from here, so I
have to subtract 3x from here. Now, I would have to
divide both sides by a negative number, right? Because I’m going to divide
both sides by negative 3. So I get negative 1/3 on this
side, and based on what we had just learned, since we’re
dividing by a negative number, we want to swap the
inequality, right? It was less than or equal,
now it’s going to be greater than or equal to x. Now, did we get the same
answer when we did it both– two different ways? Here, we got x is less than or
equal to negative 1/3, and here we got negative 1/3 is
greater than or equal to x. Well, that’s the same answer,
right? x is less than or equal to negative 1/3. And that’s– I always
find that to be the cool thing about algebra. You can tackle the problem in a
bunch of different ways, and you should always get to the
right answer as long as you, I guess, do it right. Let’s do a couple
more problems. Oh, let’s erase this thing. There you go. I’ll do a slightly harder one. Let’s say negative 8x plus 7
is greater than 5x plus 2. Let’s subtract 5x
from both sides. Negative 13x plus 7
is greater than 2. Now, we could subtract
7 from both sides. Negative 13x is
greater than minus 5. Now, we’re going to divide
both sides of this equation by negative 13. Well, very easy. It’s just x, and on this
side, negative 5 divided by negative 13 is 5/13, right? The negatives cancel out. And since we divided by a
negative, we switch the sign. x is less than 5/13. And once again, just like the
beginning, if you don’t believe me, try out some numbers. And I remember when I first
learned this, I didn’t believe the teacher, so I did try out
numbers, and that’s how I got convinced that it
actually works. When you multiply or divide
both sides of this equation by a negative sign, you
swap the inequality. And remember, that’s only when
you multiply or divide, not when you add or subtract. I think that should give
you a good idea of how to do these problems. There’s really not
much new here. You do an inequality– or I
guess you could call this an inequality equation– you do it
exactly the same way you do a normal linear equation, the
only difference being is if you multiply or you divide both
sides of the equation by a negative number, then you
swap the inequality. I think you’re ready now to
try some practice problems. Have fun.

### 100 thoughts on “Algebra: Solving Inequalities”

• September 16, 2017 at 8:44 pm

• September 29, 2017 at 10:55 pm

What app are you using to write?

• September 30, 2017 at 11:31 pm

I was 2 years old when this video came out. Also you suck middle school honors classes.

• October 2, 2017 at 11:40 pm

i hope this helps to ease my struggle in algebra1 class

• October 16, 2017 at 11:58 pm

I got little bit confused on the negative

• October 18, 2017 at 4:17 pm

so bleary

• October 19, 2017 at 3:12 am

I just wanted to watch Roblox videos

• October 20, 2017 at 3:42 pm

This music is great!

• October 31, 2017 at 12:27 pm

I don't use the method of subtracting because I wasn't taught that one (although I understand it from videos) but if I were to just change the things like:

3x = 1
Well, then I'll move the 3 to the other side and I get
X= 1/3

How does the multiplícate/divide negatives rule apply to this? (I feel like this is a dumb question )

• November 8, 2017 at 12:09 am

confused!

• November 9, 2017 at 10:20 pm

0:44

• November 11, 2017 at 4:15 am

For this inequality is it able to takeaway 2 first or the x is supposed to be done first?

• November 14, 2017 at 11:38 pm

I dont like how he doesnt explain…he kinda just tells you what to write and that is why i dont get this stuff

• November 17, 2017 at 11:58 pm

MY 7th GRADE TEACHER IS A GENIUS

• November 27, 2017 at 5:34 am

Is anyone else using this for Khan Academy?

• December 4, 2017 at 5:09 pm

Sure, this totally makes sense.

• December 7, 2017 at 10:57 pm

I suck at math

• December 13, 2017 at 6:40 pm

2017

• December 15, 2017 at 10:24 am

Khan you are the best teacher in explaining math.

• December 20, 2017 at 2:22 am

can you please do a video on how to do inequality such as h minus sixteen is less than twenty-four​?

• December 27, 2017 at 5:02 pm

Why and how did you change -5/-13 to 5/13

• January 9, 2018 at 2:46 am

Thanks help a lot 👍

• January 9, 2018 at 10:50 pm

why do we need to learn this in school?

• January 10, 2018 at 5:09 am

Thanks this makes sense how are people confused

• January 18, 2018 at 12:15 am

so lost

• January 21, 2018 at 1:20 am

Hello. I just wanna ask how do you solve it when x^2-5x+1<0?

• January 24, 2018 at 3:42 am

thank you!!

• January 24, 2018 at 12:09 pm

How do I know if I should use addition, subtraction, division etc. to solve the equation

• February 3, 2018 at 4:59 pm

6:15 Negative numbers

• February 8, 2018 at 10:41 pm

Some of this is confusing

• February 18, 2018 at 5:33 pm

I was confused but this helped me so much! thank you.

• March 20, 2018 at 1:34 pm

To make live WAY more simpler DO NOT have your variable in a negative form. Always solve so the variable is positive and then disregard the flipping of the inequality sign. That's just a pet peeve of mine.

• March 27, 2018 at 5:10 am

v. important concept in algebra

• April 2, 2018 at 1:23 pm

no clue

• April 4, 2018 at 4:55 am

#betterthanmymathteacher

• April 9, 2018 at 11:37 pm

This REALLY helps with my math homework!!! Thanks!!! =)

• April 10, 2018 at 2:08 pm

Nice explaination!!

• April 23, 2018 at 11:43 pm

For anybody who is confused basically when you divide both sides by a negative number, you swap the inequality

• May 1, 2018 at 7:58 pm

• May 6, 2018 at 7:48 pm

look at this dude i dont understand your inequalties math some people know how to do it better than you punk

• May 7, 2018 at 9:15 pm

Good sir but u explain in bit confusing way

• May 10, 2018 at 4:17 am

I can't I just want to die

• May 14, 2018 at 2:18 am

Watching this video night before the staar test (major exam in Texas)

• May 29, 2018 at 3:22 pm

5:32

i dont understand… why did the sign switched to less than?????

• June 16, 2018 at 7:38 pm

video starts at 0:00

• June 18, 2018 at 3:13 am

Lost after he drew the number line😂

• June 20, 2018 at 7:32 am

I get it

• June 22, 2018 at 12:27 am

thank you so much

• July 31, 2018 at 4:46 pm

I learn't this in 6th grade and forgot you are a live safer phew thx

• August 5, 2018 at 8:31 am

• August 11, 2018 at 9:04 am

Uhm I'm confused a little bit, why do the final answer became less than?

• August 29, 2018 at 6:14 pm

LOVE IT

• August 31, 2018 at 12:58 am

this was confusing till I replayed it and he made way to easy thx bro even more 10 years later and helps people

• September 4, 2018 at 2:24 am

I'm honestly so happy I found this. My teacher taught it in like 10 minutes for one day and I have a test on it tomorrow and it makes way more sense now

• September 6, 2018 at 8:39 pm

Thank you! helped alot

• September 13, 2018 at 12:22 pm

Currently have 5 minutes until school for a test, I definitely going to fail it. -_-

• October 10, 2018 at 2:38 am

Could u do more examples plz thx

• October 17, 2018 at 4:29 am

I love you so much 😭😭😭

• October 20, 2018 at 1:41 am

• October 21, 2018 at 4:33 am

How did you become so genius

• October 28, 2018 at 11:45 pm

Excellent content. I do see that you should change to a better quality software such as that of similar YouTubers

• October 30, 2018 at 1:50 am

I didn't get a thing he said. anyone else?

• November 4, 2018 at 12:55 pm

thank you so much i cant with math this really helped

• November 8, 2018 at 2:43 pm

OMG I FiNaLLY KNOW HOW TO DO INEQUALITIES!!! Thanks khan academy!! You are now my homeschool tutor

• November 27, 2018 at 3:06 am

OMG SO HELPFUL!!!!!! I HAVE A TEST OVER THIS TOMMOROW!!!!! THANK U!!!

• November 28, 2018 at 12:39 am

Wow everyone has there school accounts here lol

• December 3, 2018 at 9:45 pm

This is confused. 🙂

• December 7, 2018 at 5:50 pm

Good he’s my good online math tutor

• December 17, 2018 at 4:37 am

you are the bob ross of educational videos

• January 7, 2019 at 12:23 pm

I will report this tomorrow in my class 😭😭

• January 7, 2019 at 7:43 pm

So, a greater than or less than sign is essentially the same as an equal sign? But why?

• January 9, 2019 at 3:54 pm

Even though I haven’t ever heard of this topic I still understood at the end!

• January 13, 2019 at 2:10 pm

what am i supposed to do in the 3 min. part of the vid I get nothing help?

• January 24, 2019 at 11:36 pm

Still confused

• February 17, 2019 at 3:59 pm

Good, but I can't read it!

• February 20, 2019 at 4:18 pm

ummmm so at 5:14 why is the 5 negative? I keep looking to see why, but I still have no clue. I'm confused.

• March 6, 2019 at 7:08 pm

So confused

• March 8, 2019 at 2:04 am

Why do 6th graders need this ;-;

• March 22, 2019 at 2:51 am

2019 anyone

• April 12, 2019 at 12:37 am

I like these videos.

• April 24, 2019 at 8:47 pm

Thanks Khan, you reminded me what the rules were for a project

• April 28, 2019 at 11:08 pm

The problem at 5:00 how do I know which side I subtract from

• May 1, 2019 at 10:19 pm

I understood all of it (⌐□ل͜□)

• May 2, 2019 at 12:42 am

Umm after the first example complete confused.

• May 4, 2019 at 3:50 pm

Man this is so confusing these comments are killing ma😂😂😂

• May 11, 2019 at 4:15 am

12 years ago

• May 20, 2019 at 9:26 pm

Learnt until 4:03 then complete confusion

• May 22, 2019 at 11:18 pm

Enequality is my enemy

• June 15, 2019 at 6:19 pm

Khan is Legend.

• July 11, 2019 at 6:21 pm

with -8x + 7 > 5x + 2

• August 2, 2019 at 4:16 pm

Exactly what I needed

• August 3, 2019 at 9:13 am

I don't understand the last part

• August 16, 2019 at 1:38 am

Got so confused cause I thought that the greater than sign was a 7

• August 28, 2019 at 1:25 am

Marksman?

• September 24, 2019 at 10:15 pm

This would be helpful if i wasnt extremely dense

• October 11, 2019 at 3:21 am

What