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In each of the following exercises, solve the given inequality.$$25 x-x^{3} \leq 0$$

$$-5 \leq x \leq 0 \text { or } x \geq 5$$

Algebra

Chapter 0

Reviewing the Basics

Section 5

Solving Non-Linear Inequalities

Equations and Inequalities

Missouri State University

Campbell University

University of Michigan - Ann Arbor

Lectures

03:07

In each of the following e…

01:23

03:31

03:51

01:16

03:36

02:45

03:26

02:30

05:55

03:41

04:15

04:32

01:50

02:42

03:27

03:14

02:25

04:02

05:39

All right, we've got a question here. 25 x minus. X cube is less than or equal to zero and we want to solve for the inequality will start off by trying to after this 25 minus X squared, multiplied by X. And then we can try to factor this even further by saying, Well, five minus x five plus X Lesson three, which is zero. All right. Now we will set all of our factors equal to zero. So we'll start off with one. X multiply is equal to zero. Excuse me, and we'll take five. Minus X is equal to zero will solve for X, which would be then five, and then we'll have five plus X is equal to zero, which would then mean our X is equal to negative. Five. Take a number line and we're gonna set it equal from negative infinity to infinity. We plot our points of 05 and that negative five, and we will look to see whether each region will satisfy the inequality. Okay, so we'll start off with the first region, which is from five to infinity, and we'll pick the simple or just any value in between there, we could just pick six. We'll have six times five minus six and then a five plus six, and we'll set that equal. We'll solve for it, so we'll get a negative one. Time six, which is negative. Six negative six multiplied by 11 is gonna be negative. 66. We could see that negative 66 is certainly less than or equal to zero. Therefore, we can confirm that that region is satisfied. I think a value between zero and five will pick one. I'm just gonna solve. So we'll get four times. One is 44 times six is 24. We know that's positive. Values or region will not be satisfied. Value from negative 5 to 0 or negative one. We'll have five minus and negative one. The five minus one. This will turn positive. Six multiplied by a 4 24 24 times negative one is negative. 24. Therefore, this region is also satisfied. All right, let's pick a value from negative five and infinity. We could just choose negative six five minus and negative. Six would just be a positive six five minus negative. 6 to 5, plus a negative six we'll get negative one multiplied by 11, which is negative 11 times negative. Six, which is positive. 66 for this region is not satisfied. And then when we write down our X domain, we're going to say, Well, X must be greater than or equal to negative five, but less than or equal to art. Zero Well said. Greater than or equal to 95 less than 20 Hey, that's one region there and then our second region, he said, is one X is greater than equal to buy. Those will be our final answers there. I hope that clarifies the question. Thank you so much for watching.

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