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Solve 𝑥 squared minus 𝑥 minus six is less than zero.
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So, in order to solve this inequality, what we want to do first is find some critical values.
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And in order to do that, what I’m gonna do is set our inequality to an equation.
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So, what I’m gonna do is I’m gonna equate 𝑥 squared minus 𝑥 minus six to zero.
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So, what we need to do here is solve this quadratic.
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And what this is gonna give us is the two points where our quadratic crosses the 𝑥-axis.
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And the way to solve this quadratic is by using factoring.
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When we do that, we get 𝑥 minus three multiplied by 𝑥 plus two is equal to zero.
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And that’s because if we look at the two numbers that need to multiply together to give us negative six, then they’re gonna be negative three and positive two.
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We could’ve had some others, so we could’ve had negative six multiplied by one, or six multiplied by negative one, or even negative two multiplied by three.
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However, they need to sum to negative one.
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And that’s cause negative one is the coefficient of our 𝑥 term.
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Well, negative six add one is negative five.
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Six add negative one is five.
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Or negative two add three is one.
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So, none of these give negative one as a result.
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So therefore, the correct factors are the ones we found, which were negative three and two.
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And that’s because if you add two to negative three, you get negative one.
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Okay, so, we factor it now, which means that we can say that our critical value is gonna be 𝑥 equals three or 𝑥 equals negative two.
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And we got these by setting each of the parentheses equal to zero.
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And that’s because we need one of the parentheses to be equal zero because zero multiplied by anything gives the result of zero, which is what we want.
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An example of this, I’ve shown, is the right-hand parentheses because we got 𝑥 plus two is equal to zero.
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Then, subtract two from each side.
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We get 𝑥 is equal to negative two, which is what we had.
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Okay, great, we have our critical values.
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But what next?
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How do we solve our inequality?
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Well, if we sketch the quadratic, then what we’ve got is we’ve got points at negative two and three where it intersects the 𝑥-axis because this is our solutions, or our critical values.
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And we know the graph is a U-shaped parabola because our 𝑥 squared term is positive.
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And if it’s positive, then it’s a U-shaped parabola.
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If it’s negative, then it’s an n or inverted U-shaped parabola.
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Now, to solve the inequality, we need to look at the inequality sign.
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And what we’re interested in is when 𝑥 squared minus 𝑥 minus six is less than zero.
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So therefore, we’re interested in this region here, which is the region which is below the 𝑥-axis, so where 𝑦 is less than zero.
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So therefore, we can say that the inequality that would represent this is that 𝑥 is greater than negative two but less than three.
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And if we want to show this using interval notation, what we’ll have is open parentheses negative two then comma three close parentheses.
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And we use parentheses because these show that the numbers negative two and three are not included.
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And that’s cause 𝑥 is greater than negative two and less than three.
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If it was including the values, so we had 𝑥 was greater than or equal to negative two or less than or equal to three, then instead of the parentheses what we would use is brackets.