Use the technique described in Exercises 87–90 to solve each ineq...

Question

Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. x^2 - x - 6 < 0

Accepted Answer

Hey, everyone in this problem, we're asked to find the solution for the given quadratic inequality. And to express the answer in interval notation, the inequality we're given is X squared minus X minus 20 is less than zero. We're given four answer choices. Option A, the interval from negative 5 to 4. Option B, the interval from negative 4 to 5, option C the interval from negative 4 to 5 including the end points and option D, the interval from negative 5 to 4 including the end points. We're gonna start by writing out our inequality X squared minus X minus 20 is less than zero. And what we wanna do is we want to factor this equation on the left hand side first. OK? It makes it a lot easier to compare to zero when we have things multiplied together rather than adding and subtracting all of these terms. So let's rewrite our equation to factor. OK? And you can factor in any way that you like. We're gonna go ahead and do it this way and show you one method. But if you have a method that you prefer, go ahead and use that we get X squared plus four, X minus five, X minus 20 is less than zero. OK. So we're breaking up that middle term. Now, we have four terms in our equation. And what we're gonna do is factor the first two terms, we're gonna factor the second two terms. And then we're gonna combine that all together in one last step or factoring. So we have in the first two terms X squared plus four X and we have a common factor of X that we pull out. And what's left is X plus four. So we have X multiplied by X plus four in the second two terms or the sorry, the last two terms, we have negative five, X minus 20. And we have a common factor of negative five minus five in our equation multiplied by what's left over which is X plus four. So we have X multiplied by X plus four minus five, multiplied by X plus four is less than zero. And if you were to expand this, you would get back that equation we had on the previous line. OK. So you can always do that to double check your work. Now, we have this common factor of X plus four, we can factor that out. And what we're left with is X minus five. So we have X plus four, multiplied by X minus five is less than zero. And that is our factored equation. Now that it's factored, we want to look at our points of interest and, and our points of interest are gonna be where the left hand side of our equation is equal to zero. Now, if the first term is equal to zero, the entire left hand side will be equal to zero and same for the second term. So from the first term, if X plus four is equal to zero, we get that X is equal to negative four. And from the second term, if X minus five is equal to zero, then X is equal to five. So we have these two points of interest, X equals negative four and X equals positive five, we're gonna add those on to a number line and we're gonna look at the intervals on either side of these points. OK? This breaks up our number line into three distinct intervals. Now, if we have X less than negative four and we can think of say substituting X equals negative five, X plus four, our first factor is gonna be negative, it's gonna be less than zero. And the same thing for X minus five. OK? So if we look at the left hand side of our inequality, which we're gonna call L H S which is X plus four multiplied by X minus five, we have a negative multiplied by a negative and that's gonna give us a positive. So the left hand side is gonna be greater than zero. Now, our inequality, we want it to be less than zero. So greater than zero does not satisfy our inequality. So we're gonna draw a red X on this interval. It is not going to be included in our solution set. Now we move to the next interval. We have X plus four. OK? We're between negative four and five. So we can pick a test point, say X equals zero, we substitute that in X plus four is gonna be greater than zero, X minus five is still gonna be less than zero. So now we have a positive multiplied by a negative in the left hand side is gonna be less than zero. Now, that satisfies our inequality, that's exactly what we want. So we're gonna give this a green check mark to indicate that this interval is in our solution set. Now what we have to do when we have this interval is to double check the end points. So if we have X equals negative four or X equals five, the left-hand side is gonna be equal to zero. But our inequality is a strict inequality. We want this to be strictly less than zero, not less than or equal. OK? So neither of these points satisfy the inequality, we don't want to include them in our solution set. And so we're gonna have open circles at negative four and at five to indicate that we don't include those points. OK? And that's just gonna be a visual reminder when we go to write down our interval of what happens at those end points. Now, we're on our last interval, that's when X is greater than five, we have the X plus four will be greater than zero and X minus five will also be greater than zero. And so the left hand side of our equation will be positive, it's gonna be greater than zero, which does not satisfy our inequality. So this is not part of our solution. So we found that we have just one interval for this solution. And we're gonna write it in interval notation and that interval goes from negative four, up to five. And we have round brackets to indicate that we are excluding the end points. OK? Because they do not satisfy our inequality. So if we compare this to the answer choices we have, we can see that this corresponds with answer choice. B thanks everyone for watching. I hope this video helped see you in the next one.

Use the technique described in Exercises 87–90 to solve each inequality. Write the solution set in interval notation. x^2 - x - 6 < 0

Key Concepts

Inequalities

Quadratic Functions

Interval Notation

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