Quadratic inequalities: graphical approach

welcome to the presentation on quadratic inequalities before we get to quadratic inequalities let's just well 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 you just factor this quadratic X plus 3 times X minus 2 equals 0 and you would learn that the roots of this quadratic equation of this quadratic function are X is equal to minus 3 and X is equal to 2 now let's how would 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 that minus 3 y is 0 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 and we also know that the y intercept is minus 6 isn't the vertex is the y intercept or that the graph is going to look something like this and that is bumpy's what I'm drawing which I think you get the general idea if you 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 X values make this 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 in 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 -3 right or whenever X is greater than 2 right 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 solve 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 when you can 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 and 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 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 whoops minus x squared minus 3x plus 28 let me say is greater than zero 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 a little more confusing to factor so I'm going to multiply both sides so I get x squared plus 3x minus 28 and when you multiply or divide by negative with any inequality if to swap the sign so this is now going to be less than zero right and if we were to factor this we get X plus seven times X minus four is less than zero so this was equal to zero we would know that the two roots of this function let's say that let's define the function f of X let's define the function as f of X is equal to well we could find it as this or this because it's the same thing but let's just for simplicity let's define it as X plus seven times X minus four 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 seven excuse me my throat is dry it just ate too many almonds and X is equal to four now what we want to know is what X values make this inequality true if this was an equality we would be done we want to know what makes this inequality true and 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 quick 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 try x equals zero well F of zero is equal to we could do it right here F of 0 is 0 plus 7 whoops 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 what this is the function we're working with is this less than 0 well yeah it is so it actually turns out 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 you could have done it 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 I've tried it out let's say if you had try it out 5 try x equals 5 well then f of 5 would be 12 12 times 1 right which is equals to 12 F of 5 is 12 is not less than 0 no so that wouldn't have worked so once again that gives us confidence that we got the right interval and if we wanted to think about this visually all right because we got this answer when you do it visually it actually makes I think a lot of sense but maybe I'm biased let me erase this real fast if you look at it visually it looks like this oh well that's way too fat let me do it on a scan your pen if you draw it visually and this is the parabola right this is f of X the roots here are minus seven zero and four zero we're saying that for all X values between these two numbers f of X is less than zero and that looks that make sense because when is f of X less than zero well this is this is the graph of f of X right this is f of X and when is f of X less than zero 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 probably gloss out how do I know I don't include zeros well you could try it out but if you know oh 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 a 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