# How to Solve Inequalities (NancyPi)

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Published on September 28, 2018 by

MIT grad explains solving inequalities. This video focuses on solving linear inequalities. It shows when to switch the sign of the inequality, if you divide or multiply by a negative number, and is an introduction to how to solve inequalities in algebra. To skip ahead: 1) For a basic example of SOLVING AN INEQUALITY, skip to time 00:34. 2) For when you have to FLIP THE INEQUALITY SIGN when DIVIDING BY A NEGATIVE number, skip to 1:26. 3) For when you need to reverse the inequality symbol because you're MULTIPLYING BY A NEGATIVE number, skip to 2:42. The problems here are multi-step inequalities, or two-step inequalities. Nancy formerly of MathBFF explains the steps.

Algebraic equations and inequalities: what is inequality math? What does inequality mean? An inequality is a number sentence that uses an inequality symbol instead of an equality symbol (=), which means it has a greater than sign, less than sign, greater than or equal to sign, or less than or equal to sign, instead of the normal equal sign in an equation. One linear inequality definition is: an inequality that involves a linear algebraic expression and inequality signs.

How to find the solution of an inequality algebra problem:

1) A simple, BASIC LINEAR INEQUALITY example you might see on a solving inequalities worksheet or problem set is the expression 2x + 3 [less than or equal to] 11. You can solve for x just like you would for an equality, or normal equation, by trying to get x alone. This is an example of two-step inequalities, or multi-step inequalities. The first step is to subtract the constant 3 from both sides, in order to move it to the right hand side, so that you have 2x [less than or equal to] 8. The second step is to get x alone on the left side by dividing both sides by 2 so that your answer for the inequality is x [less than or equal to] 4. In this case, solving the inequality used the exact same steps you would use to solve an "equal sign" equation.

Sometimes the steps are different from solving normal math equations. Solving for x is the same as solving a linear equality (-2x + 3 = 11) except that if you ever have to divide by a negative number or multiply by a negative number while solving for x, you must flip the sign, or switch the direction of your inequality symbol less than becomes greater than, for example). Here are the two cases of inequality problems WHEN YOU NEED TO REVERSE THE SIGN of the inequality because you're multiplying or dividing both sides of the inequality by a negative number:

2) DIVIDING EXAMPLE: For an inequality like -2x + 3 [less than or equal to] 11, after moving the constant 3 to the right side, you will need to divide both sides by negative 2 in order to get x alone on the left side for the solution. WHEN YOU DIVIDE BY A NEGATIVE NUMBER, you need to reverse the direction of the inequality symbol in your expression. In this example, after you divide out -2, you get x [greater than or equal to] -4 (instead of x less than or equal to -4) for the final solution. This example was also a two-step inequality problem.

3) MULTIPLYING EXAMPLE: For examples like (-1/2)x + 4 [greater than] 7, when you've moved the constant to the right side and have the inequality (-1/2)x [greater than] 3, the fastest way to get x alone is to clear the fraction by multiplying both sides by -2. This will cancel the 2 in the denominator and remove the negative sign in front of x on the left side. But WHEN YOU MULTIPLY BY A NEGATIVE, you have to flip the direction of the inequality symbol so that your answer is x [less than] -6 instead of x [greater than] -6. This problem was also a multi-step inequality.

Note on inequality rules: you don't have to switch the symbol if you're adding a negative number or subtracting a negative number: switching the inequality sign is only for when you're multiplying or dividing by a negative number.

For quadratic inequalities examples, compound inequalities, inequality math, inequality algebra, inequality equations, inequality symbols, inequality notation, and more algebra solver, equation solver, and inequality calculator practice problems that come up on worksheets, as well as more algebra, algebra 2, and college algebra mathematics videos, check out: http://nancypi.com

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• Yay first

• plz upload the polynomial inequalities if you have it

• Very good technique

• hot stuff Miss Pi

• Thank you Nancy!

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• Yo!

• just wanna say thanks for being the best math channel on youtube! love ya nancy!

• Well I don’t love nancy

• Perfect timing! I just learned inequalities this week! Thank you so much for the extra lesson. π

• Thanks I always need to review these

• Dangit I needed this for my test today lol. Well good video anyhow thanks!

• Linear programming problem?

• Extremely thankful for all of your videos, Nancy ππΌ

• Hi @nancy,
Thanks for your help and efforts.
Can u please starts DEβS problems

• more calc 2 please π

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• and from IRAQ too.

• Your friend from saudi arabia also

• Damn Hagrid, didnβt know you started uni

• Wheres my dragon

• We waiting π₯ππ

• Please do more videos. Love from India

• I thought solution sets of inequality are expressed using set-builder notation and interval notation at the end of the answer? Is she missing something?

• Hi nancy can u do a Rational Functions episode

• Is any body watching maths? …or I am the only person.

• I am watching… what’s wrong ?

• This video will do nothing to help black people in modern day America. Thanks Obama!

• Course for a level?

• Hi Nancy. Thanks for the fantastic tuition videos. I left highschool without finishing to pursue a trade. Now I am in college doing a BEng as a mature student. Your videos have helped me out a lot over the past two years. Thank you.
Do you plan on doing a video on first order ODE?

• I remember this in an online class. Your video explained this very well. Thank you

• Can you please make a vedio on probability of indipendent events .

• Good audio. The one before had too much echo.

• Nancy is my president

• My professor has a hard Russian accent and I can’t understand anything, your videos are clutch af!

• Thanks you so much.

• NANCY PLEASE DO VECTORS PLEASEEE!! I AM BEGGING YOU! YOU ARE THE BEST MATH TEACHER EVER! LOTS OF LOVE XOX

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• You are a total teaching legend.

• Nancy i love the way you teach and especially your logarithm videos helped me a lot .Thanksπ

• You are the truth! I am in math and it is so hard. You give me the strength to keep going. On my way back to class and when I get back I will do my homework with watching you. Thank you. Have a great day

• man i hated math until i saw u Nancy , u are looking hot

• Wierdo

• Oooooof

• How old are u

• @Jayden Ng im 13

• I so wish you were my math teacher. Please keep making videos you are so easy to understand and you speak clearly and deliver the message very patiently.

• This video really helped because I have an exam next week, Thank you so much

• Very well explained, like always! Keep it up!

• thanks for helping us

• 2:24…….f the sign ??????ummmmm

• Omg weβve been on this lesson for 2 weeks and this video literally taught me

• Well, Now I’m not scared of mathematics. Thank you πππ»

• Solving inequality in Math is way easier than solving it in the Society. Thanks Nancy

• Fantastic video again from Nancy but for anyone wondering: Here is why it is flipped.

Imagine a number line. On the positive side lets say, “56 is greater than 23” (56>23). Then if we multiply both sides by -1, we essentially just change the side of zero they are on. They are still 56 and 23 away from zero, but they’re on the negative side of the number line now. (-56<-23<0<23<56) Essentially, a number larger than another number when positive, is smaller than the other number when they're both negative. Seems basic now I look at it, but didn't grasp it in the video.

• Sen484 Awesome! Thanks!

• that explained it thank you^^

• Thanks

• Wow omg thx for summarizing! I needed this since i really struggled to understand it thx!

• OMG, she explained so simply.. I was always confused how and when did the sign changed? But you made me all clear..THANKYOU

• Nice, I make similar videos. I know, nobody asked π

• Yh she really did

• Franck’s Notes hahaha your SO funny! π

• This was extremely helpful but I can’t find the compound inequality video you referred to.

• thankx mam

• Luv u from Pakistan Nancy !

• As a sociology major, I grossly misread the title.
I did learn that mathematical inequalities are much easier to solve compared to social inequality, so thanks!

• Sorry, my dude, this is a math video, not a Cultural Marxism video. For that you’ll have to see Crash Course Sociology. If you don’t have time, let me summarize: muh white man bad, women good, gays better. There you have it in a nutshell.

• @The Goodly Dragon ok but nobody asked

• @The Goodly Dragon Thank youuuuu i got a A* how did you even know that?????

• The Goodly Dragon who are you

• @The Goodly Dragon I do not mean to sound like an egotistic and presumptuous jerk but given that I graduated with honours with a BA in Sociology, I would have to say that it appears that you know little about the complexities of sociological theories. However, I apologize if my presumption is incorrect.
On a side-note, given the chaos currently occurring throughout the U.S., I would recommend taking a quick look at Das Kapital (1867) by Karl Marx, and The Permanent Revolution (1930) by Leon Trotsky.

• Where r u from

• Hi Nancy, great job! Please show each step for new comers. π

• Thanks 4 clearing my concept

• Hi, I had a question in regards to solving this inequality:

x+1/2x-3 < 1/x-3 Iβm not too sure how to solve it since we donβt know if x is positive or negative so you have to multiply the denominator by x^2

• How to Do Stuff with Ethan we donβt know if x is – or + it is an unknown

• @Marie Hill true. never mind.

• How to Do Stuff with Ethan thank you anyways! Hope youβre doing good

• This is how I did it:
(x+1)/(2x+3) < 1/(x-3) If x-3 > 0, x > 3:

(x+1)(x-3)/(2x+3) < 1 2x+3 must be positive because x > 3

(x+1)(x-3) < (2x+3) x^2 - 4 x - 6 < 0 If x-3 < 0, x < 3: (x+1)(x-3)/(2x+3) > 1 (flip the sign because x-3 is negative)

If 2x+3 < 0, 2x < -3, x < -3/2. (x+1)(x-3) < (2x+3) (flip the sign again if x 2x+3 is negative) x^2 - 4 x - 6 < 0 If 2x+3 > 0, 2x > -3, x>-3/2:

(x+1)(x-3) > (2x+3)

x^2 – 4 x – 6 < 0 So: If x > 3 then x^2 – 4 x – 6 < 0 If x < -3/2 then x^2 - 4 x - 6 < 0 If -3/2 < x < 3 then x^2 - 4x - 6 > 0

Solve the inequalities:

x^2 – 4x – 6 < 0 Complete the square: (x-2)^2 β‘ x^2 - 4x + 4 (x-2)^2 - 10 < 0 (x-2)^2 < 10 x-2 < sqrt(10) or 2-x < sqrt(10) (because 2-x and x-2 give the same squared) x-2 < sqrt(10) x < sqrt(10) + 2 or 2-x < sqrt(10) -x < sqrt(10)-2 x > 2-sqrt(10)

So:

2-sqrt(10) < x < sqrt(10) + 2 This is only true when x > 3 or x < -3/2 so If x > 3:

2-sqrt(10) < 3 so it cannot be a solution when x > 3.

So one solution is: 3 < x < sqrt(10)+2 If x < -3/2 then sqrt(10) + 2 cannot be a solution because it is more than -3/2. So another solution is: -3/2 < x < 2-sqrt(10) The solutions for x^2 - 4x - 6 > 0 are:

(x-2)^2 – 10 > 0

(x-2)^2 > 10

x-2 > sqrt(10) or

2-x > sqrt(10)

x > sqrt(10) + 2 or

-x > sqrt(10)-2
x < 2-sqrt(10) So the solutions to x^2 - 4x - 6 > 0 are:

x < 2-sqrt(10) x > sqrt(10) + 2

x^2 – 4x – 6 is only true when -3/2 < x < 3 -3/2 < 2-sqrt(10) sqrt(10) + 2 > 3

Since both have to be true, there are no values for x that satisfy the inequality when -3/2 < x < 3. So the only solutions to the original inequality (x+1)/(2x+3) < 1/(x-3) are: 3 < x < sqrt(10)+2 or -3/2 < x < 2-sqrt(10) Actually you might have meant x+(1/2x)+3 < (1/x)-3.

• She reminds me of the daughter in the Incredibles.

• oh yeah… She kind of does lol

• dude not now

• the daughter in the Incredibles is Violet

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• tanx nancy

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• I should have watched this before my test π

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• Did u pass ur grade

• At least now you know where to go in the future

• What a rare sight to see these days, an attractive girl who is also intelligent

• You must live under a rock lol

• 2:24 fu*k the sign……… π

• She said flip

• Could you stop cuzzing!

• Thank you so much Nancy! I’m at grade 7 and we’re learning this lol who else???

• I’m in 9th grade and I’m learning this lol ^^

• @Debora Tintisan me too

• year nine :/

• R u from india nancy….Nice math teacher

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• Gaandi bol Bhai π€£

• This video is super helpful thanks!

• -6

• thanks …it helped me alot

• Merci

• Thank you so much

• Thank you this is the basis for Linear Programming in my Grade XII.PLEASE DO VIDEO ON ALL TOPICS WHICH ARE BASICS FOR
CALCULUS,
VECTORS,
INVERSE TRIG,
MATRICES,
DETERMINANTS these topics

• i was scared that my parents may come from back while watching your vedio because they wouold think that i must be watching any models content

• I don’t understand why the fraction -1/2 needed to be multiplied by -2. If the fraction was -1/4 would you multiply it by -4 etc?

• LolDarrellLol Because its the -1/2 the respricol is -2/1 which = -2

• And yes

• Thanks for making this video

• Beautiful and wise ….. Best explained.

• Can u like… Be my teacher

• Such a soothing teacher.

• SO TRUUUEEE!!!!!!!!!!

I was wondering if anybody said that in the comments

• S C H O L A R

• I mean this is extremely rudimentary, wish you went in depth with quadratic inequalities

• YE I KNOOW…..

• Anyone else not focusing on the vid

• Im a bit confused. When you canceled out the 2 in the denominator, does that mean that the 1 from the fraction is a whole number or is that still part of a fraction. 2nd question is that, why do we always have to multiply by -2?

• May i go to the restroom?

• We love a math queen

• I really liked the idea of ur teaching ππππππππππ

• have a trouble with (x+3)square>(2x+3) which one is a b and c thanx in advance

• This is how I would solve it:
(x-3)^2 > 2x+3

Expand:

(x-3)^2 β‘ (x-3)(x-3) β‘ x^2-6x+9

x^2 – 6x + 9 > 2x+3

Move all terms to one side:

x^2 – 8x + 6 > 0

Complete the square:

(x-4)^2 β‘ x^2 – 8x + 16

(x-4)^2 – 10 β‘ x^2 – 8x + 6

(x-4)^2 -10 > 0

Isolate the square:

(x-4)^2 > 10

Square root both sides:

x-4 > sqrt(10)

x > sqrt(10)+4

since (x-4)^2 β‘ (4-x)^2, another solution is 4-x > sqrt(10)

4-x > sqrt(10)

-x > sqrt(10)-4

Divide both sides by -1, and flip the signs.

x < 4-sqrt(10) So the final solutions are: x > sqrt(10)+4
x < 4-sqrt(10)

• Holy thanks for the advice and help tks so much I have a big test coming with me lu k

• I love u

• Can you help me with this problem 3+9Γ>4(Γ+7) thanks

• 3+9x > 4(x+7)
3+9x > 4x+28
9x-4x > 28-3
5x > 25
x > 5

• So you subtract three for every question

• How to solve : 5(β5)^x+5 = (β5)^2x+7
So,
X=?

• Is she writing backwards so we could see it normally

• yep

• Still confuzed but less

• GUYS YOU DO REALISE THAT SHE IS WRITING BACKWARDS ON A MIRROR
DUDE THATS HARD!

• Common misconception about this channel, she writes behind a glass board and the camera is on the other side of the glass board, then the video is reversed/mirrored

• If she is a soothing teacher like to my comment

• Gracias amiga

• You are the best!

• 1:16 you got it wrong

• No, it was correct she simplified it.

• Mxxnlightx UwU yes it is she divided it therefore changes the way it points

• The way she writes 11 is weird, but I guess I shouldn’t criticize since she’s able to write backwards.

• ;0

• boooooooo

• booo

• this really help me in math on my homework thank you so much

• 0:30 π€£π€£ππ “If you don’t know what am talking about this is the right video for you”! BARS!

• I love it, she just made it easy for me.

• thank you Ms.Nancy its help me a lot, iam taking the FE exam on april 2020…

• how was that exam lol

• Josh s been cancelled

• Lol

• thank you thank youuu

• Hold up… this person is writing backwards while explaining math

What is this level of brainpower

• Daniel Kim Uh…
Maybe she used reverse filter?

• #NamcyPi You are the most beautiful teacher I’ve ever had. Thank you

• Your method of teaching is so simple and interesting. Thanks a lot from Punjab, India.

• No nigga!!

• You mentioned that you have videos for Quadratic inequalities on your channel, but I could not find them. Is that something you would consider covering?

• lol i was watching this video and my mom tried to start teaching me it backwards (aka the wrong way) this video teaches better than my mom

• idrk what grade level this is at but im still confused

• Josh s Iβm in grade 6 and weβre taking this

• Please tell me Iβm not the only one who still donβt get it!!

• Can we also divide negative half by another negative half

• I was a little surprised that a justification for reversing the inequality after multiplication by a negative was not given. I always found that that extra insight often helped me remember.

• Even though you are a math teacher
You are hot

• Hello Nancy, nice choice of content. I also have created a channel with the same content. I am Teacher Marz and I am very happy to subscribe in your channel to collaborate…

• Tq

• good nancy β€β€

• You are beautiful π

• Thank you Nancy!! Well explained

• Thanks a lot

• Hey Nancy, great video I am glad I found you here on YouTube and subscribed. I really love your style and how you present the material, very calm and clear, I need to learn how to control my breathing in my videos π I make similar content.

• I watched like 15 different inequality videos before this one and just heard “blah blah combine like terms blah blah subtract blah blah divide blah blah greater than or equal to blah”. And this 5 minute video actually made sense.

I’m telling yah, hot teacher with a nice voice makes everything easier to learn.

• Certainly doesn’t hurt. If I had more hot teachers in school, maybe I wouldn’t be where I am now!

• can u be my math teacher!!!!!!!

• Thank you you helped me so much

• Yes nancy. I hate maths but nancy makes it better

• This helped thank uuuu

• Wow amazing

• Love it

• I paid more attention to her than my math teacher.

But in serious, thanks for the step by step explanation!

• I don’t understand

• Really great and calm teaching method.πππ

• Is this a mirror

• How is she so good writing numbers backwards

• Thanks a million!!

• You didn’t say where the -2 came from at 3:17 to multiply against the fraction. Is it from the bottom part of that fraction?

• thank yo mem

• Gurl thank u so much, this helped me alot, π

• it sounds like she is bored from doing math

• Thank you

• Simple and clear explanation

• Thank you sooo much π₯°β€οΈππ»ππ»π―π