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 function f(x) = 2x^3 + 11x^2 - 7x - 6 with labeled axes.

Accepted Answer

Hey everyone welcome back in this problem. Using the following graph. Whereas to solve the given inequality, the inequality we're given is three X cubed minus five X squared is bigger than or equal to four X minus four. We can see that the function that were shown in the graph is three X cube minus five X squared minus four X plus four. Okay so we're giving information about this this graph and we have our inequality. So let's try to write our inequality to look like the function we're given in the graph so that we can compare our inequality with the information in the graph. So our inequality three X cubed minus five X squared is greater than or equal to four X minus four. Notice that if we move this for X -4 to the left hand side on the left hand side of our inequality we're going to have the function that's shown in the graph. So we get three x cubed -5 x squared -4, x plus four is greater than or equal to zero. Okay, the inequality stays the same. We're just adding and subtracting terms. So it's gonna be greater than or equal to zero. Okay, so now this function okay matches the function of the graph. We want it to be greater than or equal to zero. So we want where the graph is on or above the X axis. If the graph is on the X axis, that means that this function is equal to zero. And if the graph is above the X axis that means that this function is greater than zero. Okay, so that covers both the greater than or equal to situation. So let's look at our graph. Where is it on or above the X axis. Okay, well we see that it's on the X axis and these points at negative one at two thirds. And that too. Okay. And it's above the X axis. Kind of here or it goes up and back down and it's above the X axis on the right hand side here after it passes the X axis at two. Okay. Alright. So we've kind of outlined where the function goes above the X axis. Let's look at the corresponding X values because that's going to be our solution and the corresponding X values. Actually X values. Well, between these two points. Ok, negative one and two thirds. That's where we have this graph above the X axis or on the X axis. Okay, so we have negative one and two thirds. Okay. We need to decide if we want square brackets or round brackets. Now, for negative one, negative one has a y value of zero. Okay, it's on the X axis. We're including when this function is equal to zero. So we use a square bracket to indicate that we include negative one. Okay. Same thing with two thirds. Two thirds has a corresponding y value of zero. We want to include where this function is equal to zero. So we use a square bracket to indicate that we're including two thirds. Ok. Then we want the union. Okay. We're gonna look at the second part. So this part of the function well it's positive or above the X axis for all of these X values continuing to the right. So that means that we're going to start at two. Okay. And we're going to include two because at to the y value is zero. We're including where the function is equal to zero because we have greater than or equal to. So square brackets indicate that we include two and this is going to go all the way up to infinity and we always put a round bracket on infinity. Alright, So that is our solution there. So the inequality the solution to the inequality is going to be from negative 1 to 2 thirds included And the Union of two up to infinity. So we have solutions. See. Thanks everyone for watching. I hope this video helped see 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

Watch next

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

Watch next

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