💬 👋 We’re always here. Join our Discord to connect with other students 24/7, any time, night or day.Join Here!

Like

Numerade Educator

Like

Report

Solve Inequalities Using Multiplication and Division - Overview

In mathematics, an inequality is a conditional statement that is either true or false, depending on the value of the variable.

Topics

No Related Subtopics

Discussion

You must be signed in to discuss.
Top Educators
Anna Marie V.

Campbell University

Heather Z.

Oregon State University

Kristen K.

University of Michigan - Ann Arbor

Recommended Videos

Recommended Quiz

Algebra

Create your own quiz or take a quiz that has been automatically generated based on what you have been learning. Expose yourself to new questions and test your abilities with different levels of difficulty.

Recommended Books

Video Transcript

So in this section we're gonna be talking about how to solve inequalities using multiplication and division. So after the last section, you're probably thinking, Oh, this is just like we did before. Just do the same method because that's exactly how we solve them before. So let's see if that works here. So our first inequality says four X is greater than 20. So you probably you're thinking, Oh, we're just going to divide both sides by four because the force will cancel out. So we're left with X is greater than while 20 divided by four is five. Well, let's check to see if this works. That means any value that's larger than five. When I substituted for X in, my inequality should make a true statement. So let's try six. Well, that would mean we have four times six is greater than 20. Well, four times six is 24 24 is greater than 20. So this is true. This works so X is greater than five is, in fact, a solution. Okay, let's try the 2nd 14 x is greater than negative. 20. So you're probably thinking okay, so again, we're just going to divide both sides by four fourth cancel. So we're left with X is greater than, well, negative. 20. Divided by four is negative. Five. So again, let's go ahead and check. So that means any value that's greater than negative five should work when he substituted them. So let's try one. So we have four times. One is greater than negative. 20. Well, four times one is four and four is greater than negative 20. So this also works. Okay, so you might be thinking there's a catch because why would I put out all these examples? Well, so far, it looks like we saw just the normal way. Let's try. Number three says negative four X is greater than 20. So again, you're probably thinking Okay, we're just gonna divide both sides this time by negative four. So you have X is greater than and 20 divided by negative. Four is negative. Five. So again, this should mean we substitute than any value that's greater than negative. Five. It should make that inequality true. So let's try one again. Well, we would have negative four times. One is greater than 20. Well, negative. Four times one is negative four and negative four is not greater than 20. So this is false. That wouldn't work. So something must happen here. Well, before we figure that out, why don't we go to the last example we have Negative for X is greater than negative 20. So you're probably thinking we divide both sides by negative four because the negative force will cancel so we'll have X is greater than, well, negative. 20. Divided by negative four is positive. Five again. That means, in theory, any value greater than five should make our inequality true. So let's try six. What we would have negative four times six is greater than negative. 20. Well, negative. Four times six is negative. 24. However negative 24 is not greater than negative 20. So this doesn't actually hold true. So let's take a look. These the first two examples worked perfectly fine, and they are true. So notice when you divided both sides of your inequality by a positive value, you got the correct solution. It's when we divided both sides by a negative value that we got the incorrect solution. So let me scroll down here and let's talk about this a little further. So let me kind of put a little separator here. So remember, our first problem was negative for X is greater than 20. Now, if we were thinking about an equation, X would be equal to negative five. So here's what we have to do here. What we're going to dio is I'm going to move the negative forex term to the right hand side of our equation. So what I'm gonna dio is ad for X and you'll find after this a couple of this overview that you won't actually do this typically. Well, again, The reason why I can do this is negative for X plus four x zero. These terms would cancel. So we have Zero is greater than, well, 20 and four X or not like terms, so we would keep these separate. Now, I know we haven't talked about multi step inequalities, but you do in fact, solemn just like equations. So I would then subtract 20 from both sides of our equation. So zero minus 20 is negative. 20. So we have negative 20 is greater than four X and then to get X by itself. Sorry. Carson got away from me we would divide both sides of our equation by four. So now remember, we're divided by a positive value, which means we should get the correct result. So again, negative 20 divided by four is negative. Five we'll bring down are greater than sign, and then we'll bring down our X. Now remember, we don't usually like having the acts on the right hand side of our inequality, especially if we have to go ahead and grafted on the number line. So if we were to flip it around, we would then have X is less than negative. Five. Well, let's see if this would actually work. So that means if we pick a value for X, that's less than negative five and substituted into our inequality. It should be true. Well, a value that's less than negative. Five is negative. Six. Well, negative. Four times negative. Six. We want to know if that's greater than 20. Well, negative. Four times negative. Six is 24 and 24 isn't back greater than 20. So notice are correct. Solution should be that X is less than negative. Five. Remember before we just solved it, we had it that it was greater than so. If I scroll back, we had already solved the same exact inequality. This was number three here, so we got the same critical number. The difference is our sign flip. So that's going to be the key. When you divide both sides of an inequality by a negative, you're gonna have to flip the sign. So let's try number four again down below and see if we would get the correct solution. If we then flip the inequality side. So if we go back down here so our fourth inequality was negative for X is greater than negative 20. So we're going to divide both sides by negative four. And because we're dividing both sides by a negative, that's going to foot part inequality side. So the greater than sign will become a lesson side. Well, negative 20 divided by negative. Four is five. So let's see if five is the correct solution. So that means if we substitute any value, sorry my curse keeps running away on me. If we substitute any value, that's less than five. We'll see if this makes our inequality true. Well, let's try three. So we have negative four times three and we want to know if that's greater than negative. 20. Well, negative. Four times three is negative. 12 and negative. 12 isn't back rather than negative 20. So X is less than five is the solution. So what we figured out is when we divide both sides of our inequality by a negative, we have to flip the inequality sign. Now, notice if I scroll back up here to the top, it doesn't matter about the number on the right hand side of the equation. Kind of like a number two. The value on the right was negative, but when we divided by a positive, we still got the same answer. So the key is like examples number three and the number four. When we're dividing both sides of our inequality by a negative value, we're gonna have to flip our inequality sign. And one thing you'll find is true is this is also true for multiplication. If you multiply both sides of your inequality by a negative value, you also have to flip your inequality sign. So let's kind of make a general note here. Down at the bottom, let's scroll down one more. So General ruled that we need to make sure that we're following is if we multiply or divide both sides of our inequality by a negative value. We have to flip the inequality side. So when multiply or dividing both sides off and inequality by a negative number, you need to flip the inequality signed. So that's the key here. So again, it's in any time you multiply or divide both sides of your inequality by a negative, you need to flip the sign in order for it to be correct. Um, so in the next couple of videos, you'll see some examples, um, and see when you need to flip it when you don't need to flip it. And as you continue through the inequality unit, you'll also see this as well. But that's the key difference. So, really, when it comes to solving inequalities, your steps are the exact same as when you solve an equation where this being our only difference and the reason why it doesn't affect us when it comes to an equation is think about it. If you flipped around unequal sign, it would still be unequal sign, as opposed to when your foot in inequality sign it changes the meaning of it. So just keep this in mind