Quadratic inequalities (visual explanation) | Algebra II | Khan Academy

Quadratic inequalities (visual explanation) | Algebra II | Khan Academy


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.

74 thoughts on “Quadratic inequalities (visual explanation) | Algebra II | Khan Academy

  • July 23, 2007 at 12:14 am
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    This is helpful. Thanks.

    Reply
  • April 10, 2008 at 1:14 am
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    Excellent 5/5. Very kind of you to make these videos.

    Reply
  • March 16, 2009 at 5:10 pm
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    absolutely agree with you

    Reply
  • May 19, 2009 at 3:11 am
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    I love watching your math videos 8) !
    They're always great

    Reply
  • July 15, 2009 at 4:53 am
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    whut? whut happened??????

    Reply
  • September 4, 2009 at 4:42 am
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    thanks.

    Reply
  • October 23, 2009 at 11:53 pm
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    thanks. You pretty much own at teaching.

    Reply
  • January 27, 2010 at 6:32 am
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    thank you, sir.

    Reply
  • February 10, 2010 at 3:47 am
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    @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

    Reply
  • March 26, 2010 at 11:50 pm
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    @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.

    Reply
  • March 28, 2010 at 6:27 pm
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    negative six u mean

    Reply
  • October 18, 2010 at 8:42 pm
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    shouldnt the graph be the other way rounnd? because a<0??
    ax^2 + bx + c

    =/

    Reply
  • October 21, 2010 at 4:52 pm
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    @PLdrummer no a was -1 he multiplied the whole thing by -1 to get 1, that means ur changing the initial graph….

    Reply
  • November 23, 2010 at 10:42 am
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    @razmadaz123 rahul, go eat curry 😀

    Reply
  • December 18, 2010 at 8:20 pm
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    Weird, I was also eating almonds whilst watching this video.
    Coincidence? Almost certainly.

    Reply
  • February 6, 2011 at 10:46 am
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    LOOOOOOOOOOOOOOOOOOOOOOOOOOL, ''mty throat is dry, i was eating too many almondss'' :L:L ahahaha !

    aahhhh; Mr.Khannnnn u are wiked 🙂

    Reply
  • February 17, 2011 at 1:10 am
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    at 9:51 I shit my pants!

    Reply
  • February 26, 2011 at 2:11 pm
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    @waaathaaa The 3 and -2 are the only factors of -6 that add up to -1.

    Reply
  • April 16, 2011 at 11:22 pm
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    8:31 Thats what she said…

    Reply
  • April 29, 2011 at 2:47 am
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    @skychy lmao right

    Reply
  • May 5, 2011 at 10:37 am
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    for (2m-1)x^2-14x+1>0, why is it that b^2-4ac<0?

    Reply
  • May 14, 2011 at 4:54 am
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    @skychy wowowwowowow

    Reply
  • June 10, 2011 at 12:59 pm
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    @theillusion100 thank you for explaining that! (:

    Reply
  • August 25, 2011 at 1:39 pm
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    are you ahead if your learning this in 8th grade?

    Reply
  • September 6, 2011 at 5:22 pm
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    thank you so much! -.-

    Reply
  • October 12, 2011 at 11:01 pm
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    thanks a lot, i thought i was missing something when the text book started blabbering but you pretty much cleared up my confusion

    Reply
  • November 12, 2011 at 7:31 pm
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    @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.

    Reply
  • December 24, 2011 at 2:03 pm
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    thank you

    Reply
  • January 4, 2012 at 11:37 pm
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    shittt….. u coaughed rit to ma ears……….. i ws with the headset…… 😛 bt dt kept us live throught the vedieo… 🙂

    Reply
  • January 25, 2012 at 3:02 am
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    Thank. You. So. Much. without this channel i honestly would have NO IDEA what i am doing in trig right now.

    Reply
  • March 15, 2012 at 10:53 am
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    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?

    Reply
  • April 17, 2012 at 9:15 pm
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    thank you!!!!! U saved my life. Keep up the good work!!!!!!

    Reply
  • May 5, 2012 at 2:59 pm
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    Use the quadratic formula.

    Reply
  • June 13, 2012 at 7:52 pm
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    and khan academy once again saves my life. thank you

    Reply
  • August 20, 2012 at 10:15 am
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    better than yr 10 maths teacher

    Reply
  • September 8, 2012 at 11:52 pm
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    This actually heled me

    Reply
  • September 25, 2012 at 3:46 am
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    fuck math

    Reply
  • March 7, 2013 at 3:34 pm
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    are you the guy that created maths??

    Reply
  • March 14, 2013 at 8:26 pm
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    Thanks man, missed this lesson in class from being sick. Saved me on the exam.

    Reply
  • May 24, 2013 at 12:00 am
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    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.

    Reply
  • December 24, 2013 at 8:59 pm
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    You didn't even explain the correct way are you serious!

    Reply
  • June 3, 2014 at 6:50 am
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    thanks for this stuff, im taking my algebra EOC tomorrow and I think these videos just saved my life.

    Reply
  • September 5, 2014 at 10:08 am
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    Thanks…just explained it much better than my teacher…

    Reply
  • November 4, 2014 at 2:53 am
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    "See you in the next presentation" drops mic

    Reply
  • March 22, 2015 at 3:31 am
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    does the line have to be dotted for the less than or greater than graphs

    Reply
  • March 24, 2015 at 3:33 am
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    (I wear my earphones watching your videos) 5:57 you just blew my eardrums..damn! 

    Reply
  • April 25, 2016 at 9:31 pm
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    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.

    Reply
  • May 7, 2016 at 12:45 am
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    thank you a lot. I watch a bunch of videos but after yours all my confusions become clear

    Reply
  • August 9, 2016 at 2:33 am
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    You have improved your writing sooo much omg hahah

    Reply
  • September 28, 2016 at 11:55 pm
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    wher do u get the s and-2???

    Reply
  • October 22, 2016 at 9:33 am
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    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
    MAGIC

    Reply
  • February 16, 2017 at 3:23 am
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    This has been the best explanation and I finally understand how to do this! THANK YOU!!!

    Reply
  • July 1, 2017 at 4:38 am
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    please upload modulus quadratic questions

    Reply
  • October 16, 2017 at 3:04 am
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    I can hear your saliva when you talk.

    Reply
  • October 22, 2017 at 4:35 am
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    honestly your vids help me so much i love how you cut the crap and teach it as it is like ur the best

    Reply
  • November 30, 2017 at 3:22 pm
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    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.

    Reply
  • December 11, 2017 at 12:55 am
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    DARRRRRN YOOOOOOOU ALMONDS!

    Reply
  • December 11, 2017 at 12:59 am
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    8:32 lines have feelings too!

    Reply
  • February 15, 2018 at 8:44 pm
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    Who is here in 2018? His voice is very young compared to now. 🙂

    Reply
  • February 22, 2018 at 12:36 pm
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    How do get a solution for an inequality without roots

    Reply
  • May 13, 2018 at 3:36 am
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    i see most of the people making negative comments. y'all so salty. This video just saved my couple of marks!

    Reply
  • May 18, 2018 at 8:56 am
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    Why if you multiplied the whole expression,the inequalities notation changes?

    Reply
  • June 2, 2018 at 5:15 pm
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    Finally, now i can rest in peace.

    Reply
  • July 12, 2018 at 12:22 am
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    8:40 that graph makes me sick

    Reply
  • July 18, 2018 at 5:22 pm
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    Thank you

    Reply
  • August 24, 2018 at 1:27 pm
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    Imong mama

    Reply
  • October 7, 2018 at 8:49 pm
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    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.

    Reply
  • November 8, 2018 at 4:49 am
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    I really enjoy your videos. Maybe remake this one though 😀

    Reply
  • November 14, 2018 at 4:07 am
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    cough sorry my throats really dry I just ate too many almonds 😂

    Reply
  • April 10, 2019 at 1:50 am
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    Wait so im confused howw is -28 between -7 and 4?

    Reply
  • May 16, 2019 at 10:47 pm
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    Wait but if the coefficient on x is negative then shouldn't it be an inverse parabola?

    Reply
  • August 1, 2019 at 3:49 am
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    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?

    Reply
  • October 8, 2019 at 1:12 am
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    I was literally having a breakdown looking for any sort of explanation and this nailed it, thank you

    Reply
  • October 17, 2019 at 4:39 am
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    thx but u can stop to dirnk water if u really need

    Reply

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