Solve each polynomial inequality in Exercises 1–42 and graph the ...

Question

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x^3≤4x^2

Accepted Answer

Welcome back. I am so glad you're here. We are given this inequality X cubed is less than or equal to seven X squared. And we are asked to solve it. The first step in solving this inequality is we want to get one of the sides of our inequality equal to zero. I'm going to subtract seven X squared from both sides. So that the term with the higher degree which is X cubed is still positive. So we've got X cubed minus seven X squared is less than or equal to zero. And now the next step in solving our inequality is that we want to find our zeros. We want to find where this sets intervals on our number line. So we're going to set this equal to zero will set X cubed minus seven X squared equal to zero. So we can find out where X is equal to zero. So now we can solve this by factoring out an X squared, we can divide both terms by X squared, X cubed, divided by X squared is X negative X squared, divided by X squared is negative seven. Now we've got two factors that we can set, both of them equal to zero, X squared equals zero and X minus seven equals zero. For the first one, we can take the square root of both sides. And we've got X on the left hand side equals plus or minus the square root of zero. Well, that's just zero. For the second one, X minus seven equals zero. We can add seven to both sides and we've got X equals a positive seven. So on a number line, those would be our two intervals. That's not the straightest number line, but you get the idea. So we can set a zero and a seven. And now what we want to find out is where on the number line is this inequality true. Where will we find values that make X cubed is less than, or equal to seven X squared true. So how we figure that out as we choose test points, we're going to choose test points in each of these intervals. We'll choose one less than zero. I'm going to choose negative one. Well, choose one between zero and seven. I'm going to choose one. I'm not putting these where they would go on the number line and we'll choose one greater than seven, which is eight. And now I'm going to plug each of those test points into our original inequality and see where it's true. So instead of X cubed, it's going to be a negative one cubed. They're getting plugged in for X is less than or equal to seven times not X squared. But now negative one squared, negative one cubed is still negative one less than or equal to seven times negative one squared is positive one. So we've got negative one is less than or equal to seven times one which is seven. Is that true? Is negative one less than or equal to seven? Yes, it is. So this inequality is true for numbers less than zero. Let's find out about numbers in between zero and seven trying one. We've got one cubed is less than, or equal to seven times one squared. One cubed is one less than or equal to seven times one squared is 11 is less than, or equal to seven times one which is seven. Is that true? Is one less than, or equal to seven? Yes. So it's also true for numbers between zero and seven. Now let's try eight. We've got eight cubed is less than, or equal to seven times eight squared. Eight cubed is 512 is less than, or equal to seven times eight squared, which is 64 512 is less than or equal to seven times 64. 7 times 64 is 448. Is this true? Is 512 less than or equal to 448. Nope, it is not true. Okay. We've figured out what's happening between our intervals. How about at our intervals what's happening at zero? And at seven, we look back at our original inequality and we see this is not strict, it's less than or equal to. So zero and seven are included for this inequality. So that means that from seven all the way down to negative infinity, I draw a solid line to show that it includes seven, all the way down to negative infinity. That is where inequality lies. How would we write this in interval notation? We put negative infinity after parentheses. Comma 72 shows all the numbers from negative infinity up to seven and including seven. So that gives us a bracket. Alright. We look at our answer choices and this matches with answer choice. A well done. We'll catch you on the next one.

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x^3≤4x^2

Key Concepts

Polynomial Inequalities

Interval Notation

Graphing Solutions on a Number Line

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