Welcome to the presentation

on quadratic inequalities. Before we get to quadratic

inequalities, let’s just start graphing some functions and

interpret them and then we’ll slowly move to the

inequalities. Let’s say I had f of x is equal

to x squared plus x minus 6. Well, if we wanted to figure

out where this function intersects the x-axis or the

roots of it, we learned in our factoring quadratics that we

could just set f of x is equal to 0, right? Because f of x equals 0 when

you’re intersecting the x-axis. So you would say x squared

plus x minus 6 is equal to 0. And you just factor

this quadratic. x plus 3 times x

minus 2 equals 0. And you would learn that the

roots of this quadratic function are x is equal to

minus 3, and x is equal to 2. How would we visualize this? Well let’s draw this

quadratic function. Those are my very uneven lines. So the roots are x is

equal to negative 3. So this is, right here, x is at

minus 3y0 — by definition one of the roots is where

f of x is equal to 0. So the y, or the f of

x axis here is 0. The coordinate is 0. And this point here

is 2 comma 0. Once again, this is the x-axis,

and this is the f of x-axis. We also know that the y

intercept is minus 6. This isn’t the vertex,

this is the y intercept. And that the graph is going to

look something like this — not as bumpy as what I’m drawing,

which I think you get the general idea if you’ve ever

seen a clean parabola. It looks like that with x minus

3 here, and x is 2 here. Pretty straightforward. We figured out the roots, we

figured out what it looks like. Now what if we, instead of

wanting to know where f of x is equal to 0, which is these two

points, what if we wanted to know where f of x

is greater than 0? What x values make f

of x greater than 0? Or another way of saying

it, what values make the statement true? x squared plus x minus 6

is greater than 0, Right, this is just f of x. Well if we look at the graph,

when is f of x greater than 0? Well this is the f of x

axis, and when are we in positive territory? Well f of x is greater than

0 here — let me draw that another color — is greater

than 0 here, right? Because it’s above the x-axis. And f of x is greater

than 0 here. So just visually looking at it,

what x values make this true? Well, this is true whenever x

is less than minus 3, right, or whenever x is greater than 2. Because when x is greater than

2, f of x is greater than 0, and when x is less than

negative 3, f of x is greater than 0. So we would say the solution to

this quadratic inequality, and we pretty much solved this

visually, is x is less than minus 3, or x is

greater than 2. And you could test it out. You could try out the number

minus 4, and you should get f of x being greater than 0. You could try it out here. Or you could try the number 3

and make sure that this works. And you can just make sure

that, you could, for example, try out the number 0 and make

sure that 0 doesn’t work, right, because 0 is

between the two roots. It actually turns out that

when x is equal to 0, f of x is minus 6, which is

definitely less than 0. So I think this will give you a

visual intuition of what this quadratic inequality means. Now with that visual intuition

in the back of your mind, let’s do some more problems and maybe

we won’t have to go through the exercise of drawing it, but

maybe I will draw it just to make sure that the

point hits home. Let me give you a slightly

trickier problem. Let’s say I had minus x squared

minus 3x plus 28, let me say, is greater than 0. Well I want to get rid of

this negative sign in front of the x squared. I just don’t like it there

because it makes it look more confusing to factor. I’m going to multiply

everything by negative 1. Both sides. I get x squared plus 3x minus

28, and when you multiply or divide by a negative, with any

inequality you have to swap the sign. So this is now going

to be less than 0. And if we were to factor this,

we get x plus 7 times x minus 4 is less than 0. So if this was equal to 0, we

would know that the two roots of this function — let’s

define the function f of x — let’s define the function as f

of x is equal to — well we can define it as this or this

because they’re the same thing. But for simplicity let’s define

it as x plus 7 times x minus 4. That’s f of x, right? Well, after factoring it, we

know that the roots of this, the roots are x is equal to

minus 7, and x is equal to 4. Now what we want to know

is what x values make this inequality true? If this was any

equality we’d be done. But we want to know what

makes this inequality true. I’ll give you a little bit of a

trick, it’s always going to be the numbers in between the

two roots or outside of the two roots. So what I do whenever I’m doing

this on a test or something, I just test numbers that are

either between the roots or outside of the two roots. So let’s pick a number that’s

between x equals minus 7 and x equals 4. Well let’s try x equals 0. Well, f of 0 is equal to — we

could do it right here — f of 0 is 0 plus 7 times 0 minus 4

is just 7 times minus 4, which is minus 28. So f of 0 is minus 28. Now is this — this is the

function we’re working with — is this less than 0? Well yeah, it is. So it actually turns that a

number, an x value between the two roots works. So actually I immediately

know that the answer here is all of the x’s that are

between the two roots. So we could say that the

solution to this is minus 7 is less than x

which is less than 4. Because now the other way. You could have tried a number

that’s outside of the roots, either less than minus 7 or

greater than 4 and have tried it out. Let’s say if you

had tried out 5. Try x equals 5. Well then f of 5 would

be 12 times 1, right, which is equal to 12. f of 5 is 12. Is that less than 0? No. So that wouldn’t have worked. So once again, that gives

us a confidence that we got the right interval. And if we wanted to think about

this visually, because we got this answer, when you do it

visually it actually makes, I think, a lot of sense,

but maybe I’m biased. If you look at it visually

it looks like this. If you drive visually and this

is the parabola, this is f of x, the roots here are minus 7,

0 and 4, 0, we’re saying that for all x values between these

two numbers, f of x is less than 0. And that makes sense, because

when is f of x less than 0? Well this is the

graph of f of x. And when is f of x less than 0? Right here. So what x values give us that? Well the x values that give

us that are right here. I hope I’m not confusing

you too much with these visual graphs. And you’re probably saying,

well how do I know I don’t include 0? Well you could try it out, but

if you — oh, well how come I don’t include the roots? Well at the roots, f

of x is equal to 0. So if this was this, if this

was less than or equal to 0, then the answer would be

negative 7 is less than or equal to x is less

than or equal to 4. I hope that gives you a sense. You pretty much just have to

try number in between the roots, and try number outside

of the roots, and that tells you what interval will

make the inequality true. I’ll see you in the

next presentation.

This is helpful. Thanks.

Excellent 5/5. Very kind of you to make these videos.

absolutely agree with you

I love watching your math videos 8) !

They're always great

whut? whut happened??????

thanks.

thanks. You pretty much own at teaching.

thank you, sir.

@1080portal because when you factor you set the two equal to 0. like x+3= 0 and x-2 = 0 and then solve. hope i helped

@digforbear The min point is necessary for this, but not by finding the derivative. the point is to know that the x-coordinate of the vertex is -b/2a, hence vertex is (-.5,-6.25) and not (0,-6) as stated in the video. The comment 'the vertex is the y-intercept' is incorrect and rarely the case. These videos are great to see but please be aware minor errors may exist.

negative six u mean

shouldnt the graph be the other way rounnd? because a<0??

ax^2 + bx + c

=/

@PLdrummer no a was -1 he multiplied the whole thing by -1 to get 1, that means ur changing the initial graph….

@razmadaz123 rahul, go eat curry ðŸ˜€

Weird, I was also eating almonds whilst watching this video.

Coincidence? Almost certainly.

LOOOOOOOOOOOOOOOOOOOOOOOOOOL, ''mty throat is dry, i was eating too many almondss'' :L:L ahahaha !

aahhhh; Mr.Khannnnn u are wiked ðŸ™‚

at 9:51 I shit my pants!

@waaathaaa The 3 and -2 are the only factors of -6 that add up to -1.

8:31 Thats what she said…

@skychy lmao right

for (2m-1)x^2-14x+1>0, why is it that b^2-4ac<0?

@skychy wowowwowowow

@theillusion100 thank you for explaining that! (:

are you ahead if your learning this in 8th grade?

thank you so much! -.-

thanks a lot, i thought i was missing something when the text book started blabbering but you pretty much cleared up my confusion

@Ileong862 a little bit yes. it just depends a lot on the high school your going to. people go at their own pace i know juniors and even seniors that are barely learning this stuff.

thank you

shittt….. u coaughed rit to ma ears……….. i ws with the headset…… ðŸ˜› bt dt kept us live throught the vedieo… ðŸ™‚

Thank. You. So. Much. without this channel i honestly would have NO IDEA what i am doing in trig right now.

considering the second example

say you didn't know how to do it visually. don't you have to test all in intervals? here you only tested values x>4

while i'm at it, can you work a non-factorable quadratic inequality using a non-graphical method?

thank you!!!!! U saved my life. Keep up the good work!!!!!!

Use the quadratic formula.

and khan academy once again saves my life. thank you

better than yr 10 maths teacher

This actually heled me

fuck math

are you the guy that created maths??

Thanks man, missed this lesson in class from being sick. Saved me on the exam.

Well… solving >0 for F(-x) is numerically the same as <0 for F(x); notice how he and I are switching the more-than sign for a less-than sign because of multiplication by -1, so it is kept numerically equivalent. This is a basic principle of inequalities.

If he restored the function back to F(-x) as you call it, graphically you would get an upside down parabola, but instead have to look for values that gave you positive evaluations. It is the exact same range that solve either case.

You didn't even explain the correct way are you serious!

thanks for this stuff, im taking my algebra EOC tomorrow and I think these videos just saved my life.

Thanks…just explained it much better than my teacher…

"See you in the next presentation"

drops micdoes the line have to be dotted for the less than or greater than graphs

(I wear my earphones watching your videos) 5:57 you just blew my eardrums..damn!Â

If someone needs extra help use this app: Calculator Board.

https://play.google.com/store/apps/details?id=com.appiscooll.freemagicboardmath

That app solves equations and demonstrates how it was done.

thank you a lot. I watch a bunch of videos but after yours all my confusions become clear

You have improved your writing sooo much omg hahah

wher do u get the s and-2???

well here's an idea. you get two roots. put the smaller root on the left then the original inequality sign. then x then the sign again and then bigger number. works everytime.

for example. for x2+3x-28<0

the roots are -7 and 4.

so

-7 < x < 4

MAGICThis has been the best explanation and I finally understand how to do this! THANK YOU!!!

please upload modulus quadratic questions

I can hear your saliva when you talk.

honestly your vids help me so much i love how you cut the crap and teach it as it is like ur the best

Please for the love of god. pause the video. clear your throat. the last half of this was TERRIBLE to listen to your mouth/throat noises.

DARRRRRN YOOOOOOOU ALMONDS!

8:32 lines have feelings too!

Who is here in 2018? His voice is very young compared to now. ðŸ™‚

How do get a solution for an inequality without roots

i see most of the people making negative comments. y'all so salty. This video just saved my couple of marks!

Why if you multiplied the whole expression,the inequalities notation changes?

Finally, now i can rest in peace.

8:40 that graph makes me sick

Thank you

Imong mama

I'm sorry, but you don't really explain why you do each step, you just tell us to do certain things, and if you don't really have a good understanding of the subject, you finish the video more confused than you started.

I really enjoy your videos. Maybe remake this one though ðŸ˜€

coughsorry my throats really dry I just ate too many almonds ðŸ˜‚Wait so im confused howw is -28 between -7 and 4?

Wait but if the coefficient on x is negative then shouldn't it be an inverse parabola?

Is this method (using a graph) the same as using the sign charts method? But like just different styles? Will i get the same outcome or results if i use the sign charts instead?

I was literally having a breakdown looking for any sort of explanation and this nailed it,

thank youthx but u can stop to dirnk water if u really need