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How To Solve Quadratic Inequalities
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Mar 2, 2025
In our last session, we learned how to graph quadratic inequalities, today we'll learn how to solve quadratic inequalities.
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0:01
all right so now let me ask question
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right look what if I had changed this
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look here real quick because you'll
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understand it what if I change this to
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what to greater than zero right how do
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you change it where do I want to be if I
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change it it wasn't always greater than
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if I want it to be greater than zero
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what interval am I choosing this time oh
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it would
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be7 what negative Infinity infity to
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what6 and what
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positive infinity and it's still open
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open parentheses right it's going to be
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negative Infinity to -6 and then one to
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positive Infinity now the reason why
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still open parth because it's not more
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than or equal okay so if it has the line
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underneath brackets yeah when the line
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is underneath if I change now this to
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this now you going to have close here
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and close here but that's not what I
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have right is strictly positive and
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strictly negative so that means these
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two Corner Point cannot be included that
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make sense wait you go your eyes like
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this processing all right I like the
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processing is that yes sir can we do the
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number line in our head or have you can
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do it in your head if you trust your
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head exactly so draw
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it so if it's less than or equal to or
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greater than equal to it's got
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brackets yeah let's do another problem
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here I have x² right + 3x +
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10 is less than or equal to zero now
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what should automatically go off in your
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head is that it's going to be a bracket
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right so now we just need to find the
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solutions right now let's look at it so
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I'm going to rewrite this I'm going to
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first say I'm going to let x^2 + 3x + 10
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equal to zero right now the problem is
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this I have a negative here I don't like
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that so I'm I'm going to try to factor
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factor it right so if I put -1 out here
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it gives me X2 - 3x - 10 is equal to Z
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now it's easier for me to factor right I
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want two numbers whose product is
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what3 10 but then add of to so what are
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is5 and 2 right 5 and 2 so be -1 * x - 5
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and then what x + 2 does that make sense
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to everybody have we got this right and
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now we're going to set it up set each
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one to zero right so we got now I can
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disregard ative 1 so x - 5 is equal 0
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or x + 2 is equal Z so X will be either
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equal to what 5 or X will be equal to
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you just IGN why I just IGN the negative
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yeah I can because it multiplies numb so
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it doesn't matter right it won't change
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anything here here I can ignore that so
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now unlike Nathan oh no Nicholas I don't
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have a beautiful mind like that I need
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to draw my number like cuz I can't
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figure things like in my head I'm old by
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the way
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so page right so again now we need to
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choose some test points right is it
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stopping again I don't know
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notification set low power oh don't
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worry about that yeah that's fine my all
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right
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so I'm going to choose again choose the
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number between -2 and
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5 oh wait Z zero right why am I always
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end up with the worst
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markers right so these are called by the
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way these are called test points right
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zero right a number after five six right
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a
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number24 so we going to we going to take
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all these and now we're going to plug it
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in here
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right yeah you can
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do3 no don't don't let it be happy right
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and then now watch here I'm going to
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plug him in here to see what happens
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right so I have -1 that's already
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outside time what x so -3 - 5 * -3 + 2
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right that's what I have if I plug it in
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here I get1 * 8 * 1 I got three
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negatives so that me
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what8 yeah cancel they cancel out that
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gives me what8 right it gives me8 which
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is what less than zero right now what if
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I pluging zero I'm going back here 1 *
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what 0 - 5 and then 0 + 2 right that
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gives me
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-1 * 5 which is POS 5 * 2 that's a
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positive right it gives me what 10 10
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which is more than Z my
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voice and then lastly I put in 6 right
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so it's 1 * 6 - 5 and then 6 + 2 so that
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is -1 * 1 * uh 8 which is uh8 which is
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less than zero right but guess what I
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have it here right this is uh when you
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plug3 it was what Less Than Zero when we
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plug in uh z z it was more than zero and
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when we plug in this it was less than
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zero but the problem one wants me to be
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what less than or equal to zero so what
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am I going to be I'm going to be where
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anybody tell me I'm confused I want to
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be how is2 more than zero I'm not saying
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is more than Z if I'm if here when I
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plug3 right plug3 it was less than zero
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is your answer um m
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six to positive infity where do you see
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six I mean my test Point yeah my test
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points are here I'm D my test point I
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can erase them right so I want to be
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what I want to be less than or equal to
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zero so I'm going to be what from
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negative Infinity to
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what -2 bracket or parentheses and then
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what and then to five to positive
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Infinity right so it's a union of these
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two sets that's going to be my my
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solution right because this is where I'm
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than zero how do you get one solution
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always well it depends it all depends on
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the question here since this is less or
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equal to zero I'm just looking at where
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this is happening it happens here and it
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happens here I am not worried about this
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one so I want to be between negative
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Infinity to -2 and then five to positive
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Infinity is anybody having trouble with
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this we going to do a lot of problems so
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that we make sense you can close that up
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now we just want to spend the time doing
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a lot of work