In Exercises 71–72, use the graph of the polynomial function to s...

Question

In Exercises 71–72, use the graph of the polynomial function to solve each inequality.

Graph of a polynomial function showing roots at (-1,0), (2/3,0), and (2,0).

2x^3 + 11x^2 < 7x + 6

Graph of the polynomial function f(x) = 2x^3 + 11x^2 - 7x - 6.

Accepted Answer

Hey everyone in this problem. Using the following graph for us to solve the given inequality. The inequality we're given is three X cubed minus five X squared is less than four X minus four. Okay, now if we look at the graph we're given, you'll see that the graph is of three X cubed minus five X squared minus four X plus four. So our inequality three X cubed minus five X squared less than four X minus four. Okay, now we want to find a way to write our inequality. Okay, so it has a function that looks like the function that this graph is. Okay, so there's three X cubed minus five X squared minus four, X plus four. If we can do that then we can compare inequality with the actual graph. So we see that if we can move this four x minus four to the left hand side, we will have this function on the left hand side. So let's do that. We get three X cubed -5 x squared -4 x plus four. And that's going to be less than zero. We keep that sign the same when we're just adding and subtracting terms. Alright, so now we have this function okay, Which is the same as this function here and we want to know when it is less than zero. Okay, so we want to find what when function is Okay if it's less than zero, that means it's below the X axis. So we have our X axis running here. We want to find when the function is below the X axis. Well the function is below the X axis here. Okay. And it's below the X axis here. So if we look at the corresponding X values when it is below that X axis. Okay. We see that. It's any X values from here onwards kind of here. Over. Yeah. So here is that negative 1? Okay. And then we see that we also have the X values between these two points. Okay, so between one almost one and two. Oh sorry, two thirds were given that the problem? Two thirds and two. Huh? So those are the X values where we have this function below the X axis meaning that that function is less than zero and that satisfies our inequality. Okay, so let's just write up these intervals the interval on the left. Okay. It goes all the way over to negative infinity. Okay, This function keeps going down down down we go from negative infinity to this point negative one. Ok? And notice that we have less than zero but not less than or equal to zero. Okay, We don't have less than or equal to zero. So we don't want to include the point negative one because that negative one. This function is equal to zero. So we're gonna do a round brackets, indicate that we're not including the point negative one. Now we're gonna have a union of the other interval that's included as well and this interval goes from 2/3 Up to two. And again we're using round brackets, because at 2/3 and act to the function is equal to zero and we only have less than zero. We don't have a less than or equal to zero in our inequality. And so that is going to be our answer. And we have the union of the intervals from negative infinity to negative one and from two thirds to two. So we have answer A here. Thanks everyone for watching. I hope this video helps you in the next one.

In Exercises 71–72, use the graph of the polynomial function to solve each inequality.
2x^3 + 11x^2 < 7x + 6

Key Concepts

Polynomial Functions

Graphing Inequalities

Roots and Intervals

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