Solving Quadratic Inequalities
Okay in this video I’m going to talk about solving quadratic inequalities and this will be the first example and the basic idea is you make your inequality into an equation you then find the solutions of that by either completing the square you can factor it you can use the quadratic formula and from there we’ll make a number line.
Putting our solutions on the number line we’ll check those solutions in.
The inequality as well as a point from each interval okay so suppose we have x squared plus 2x minus 8 okay I’m going to factor this well first we’ll turn it into an equation so x squared plus 2x minus 8 will just make it equal to 0 ok that doesn’t matter what your inequality is just make it equal to.
0 and then I think we can factor this one so let’s see we’ll have X and X we need two.
Numbers that multiply to negative 8 but add up to positive 2 well I think positive 4 and negative 2 would do that for us so we’ll get negative 8 and then we’ll get our positive 2 in the middle equals 0 so then we set each piece equal to 0 so we’ll get X plus 4 equals 0 or x equals negative 4 we’ll do the same thing with the other part X minus 2.
Equals 0 so if we add we’ll simply get x equals 2 alright so now.
I’m going to make my number line okay I’m going to put these points on there I’m going to put negative 4 down and then I’m.
Going to put positive 2 on there and those are the only numbers I’m going to put on my number line okay so let’s look at our inequality.
And you can even factor your inequality which would be the exact same thing we would have X plus 4 and then X minus 2 greater than or equal.
To zero okay and I like to think about it and its factored form I think it just makes the arithmetic a little easier honestly okay so we have to check each point so well let’s do that so we have to check negative 4 but again if I check negative 4 use a different color maybe one that’ll show up a little better so we have to check negative 4 but notice if I plug.
Negative 4 into my inequality it would be equivalent to plugging it into the the second second line here so we would get negative 4 plus 4 which is going to be 0 negative 4.
Minus 2 well is that greater than or equal to 0 well on the first part you get 0 well 0 times anything is 0 so certainly 0 is greater than or equal to 0 so I’m going to shade it in to indicate that that point works and the same thing if you plug in 2 you know from our first what we did at the beginning we basically said we figured out what values give you 0 if you plug.
2 in we’re going to get 0 out and again 0 is greater than or equal to 0 so if your inequality is greater than equal to.
Or less than equal to the solutions will always work ok because those are the things in particular that give you.
To check a number from each interval we have to check a number smaller than negative 4 in between negative 4 & 2 and.
Then we’ll pick a number bigger than 2 pick any number that you want so maybe I’ll plug in x equals negative 10 and I’ll check that so I’ll get.
Negative 10 plus 4 and then I’ll get negative 10 minus 2 and I’m asking myself is this.
Greater than or equal to zero if it is that means every.
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