Quadratic inequalities (visual explanation) How to solve a quadratic inequality. Visual intuition of what a quadratic inequality means.
Quadratic inequalities (visual explanation)
- 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?
- 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 draw it 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.
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