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the question asked us to solve this inequalities. We
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have two x cubed minus X squared, and that
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is less the nine minus 18 X. So what
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I'm gonna do first is take everything to one side
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. So to x cubed minus X squared and then
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plus 18 x minus nine, it is less than
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zero. They're gonna try to factor. So I
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got my acts on the 18. So I'm gonna
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factor by grouping and benefactor, these two. I'm
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gonna take out an X squared and I'm gonna be
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left with two X minus one. And it needs
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to I'm gonna take out a common factor of nine
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, and I'm gonna be laughing, too. X
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minus one is less than zero. So that gives
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me X squared. Plus nine multiplied by two X
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minus. One is less than zero. So that's
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how it factors. This doesn't factor any further.
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So we actually get X squared is equal to minus
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nine. We say that's not possible. So there's
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no solution there because it would mean that something squared
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is negative and that just does not happen. And
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the other option is that two x minus one is
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equal to zero which means that two X is equal
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to one, which means X is 1/2. So
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our only real concern is the 1/2. So we're
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gonna have a very simple number line where we just
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have 1/2 and I just need to know what the
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number before 1/2. So let's plug in zero.
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Let's say what happens when x zero and all we're
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looking at is the X squared plus nine and the
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two X minus one. And then the combination of
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that. Okay, so you don't have to necessarily
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write this out every time. I'm just gonna show
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you what we do. So when I put in
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zero, I get positive. Nine zero minus one
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is negative, Which means this is negative. So
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before that, 1/2 its negative and then you need
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something bigger than that. So I'm gonna say when
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X is one, I get one squared plus nines
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. That's still positive. Uh, two times one
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minus one is positive one. So this is positive
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. So it's positive after, So I want to
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know what it's less than zero. So my solution
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is the negative part. So my solution is the
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interval from minus infinity up to 1/2 because that is
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where it's negative