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Welcome to Algebra I: Properties of Inequality

18 minutes

Hey, guys, welcome to Algebra I. Today's lesson will focus on the properties of inequality. Back when we were solving equations, we spent time covering the properties of equality. Now we're moving on to inequalities and the properties that deal with those problems. Ready? Let's go.

(Describer) She gets a stylus and presses buttons on a screen.

Okay. Before we jump into inequalities, let's think back for a bit and make sure we're familiar with all of the symbols for inequality. 'Cause we know for inequality, it means that there's a value that's less than or greater than something, or greater than or equal to, or not equal to. There's a lot of symbols. For this one, the less than symbol--

(Describer) ...like the end of an arrow pointing left.

Okay. And a trick to keeping straight less than and greater than is if you were to expand... Back in first grade, we'd call it "the alligator mouth." If you opened it up, it looks like an L. If you look at that little alligator mouth, that's also the left, looks like a little L. So, that's less than. Okay. This symbol, when I have the little line underneath it, that is less than or equal to.

(Describer) She writes beside the symbol.

Okay. Then this one, that was greater than.

(Describer) ...like the end of an arrow pointing right.

Before we move on, let's play around with this. So, greater than, for example. If you had an inequality that said x was greater than five... Let's think about some solutions to this. Basically any number that's greater than five would make this inequality true. So, six, that would work. Seven would work. 1,200 would work. 75 million would work. Would five work? No. Because five is not actually greater than five. Would two work? No. Because two is not greater than five. This is just a way of saying x is greater than some value. I want to jog your memory with what inequalities meant while we tackle reviewing these symbols of inequality. Let's keep moving through. Here, greater than or equal to.

(Describer) It's a greater than symbol with the short horizontal line under it.

Okay. And the last one, if you run into that inequality symbol, that is "not equal to."

(Describer) An equals sign with a diagonal line through it.

You don't run into this one too often, but every now and again, it will pop up in Algebra I. Let's keep moving on to the properties. Okay, so, the transitive property of inequality. For all of these properties, we are going to assume that a, b, and c are real numbers. Let's interpret this. If a is less than b, and b is less than c, then the transitive property says that a is less then c. Let me tell you what I mean. Let's say a is 1, b is 2, and c is 3. Let's just assign them some values. What this is saying is if a is less than b-- if 1 is less than 2-- and b is less than c-- so, and 2 is less than 3-- then a is less than c. Then is 1 less than 3. It's a play off of the chain rule. If 1 is less than 2, and 2 is less than 3, then 1 is less than 3. That's the transitive property of inequality. The same thing is true if we are working with greater than. If a is greater than b, and b is greater than c, then a is greater than c. Same thing is true. Let's keep going. Addition property of inequality. Okay. If a is less than b, then a plus c will be less then b plus c. Let me show you what I mean. Let's just stick with a is 1, b is 2, and c is 3 for this case. This is saying, "If a is less than b." If 1 is less than 2, then a plus c-- then 1 plus 3-- is less than b plus c-- 2 plus 3. Let's see if that's true. 1 plus 3, that's 4. 2 plus 3, that's 5. Okay. So, that's true. It's saying that if you have two values, and one is smaller than the other, if you add the same quantity to each value, that same relationship is still true. Whatever number was smaller before you added something to it is still gonna be smaller after you add something to both sides. That's all the addition property of inequality is saying. Same thing is true for greater than. If a is greater than b, then a plus c is greater than b plus c. All right, let's keep going. Subtraction property of inequality. All right. If a is less than b-- you probably know what I'm about to write down-- then a minus c... is less than b minus c. Okay? Let's play around with some numbers here. I was gonna try to avoid negative numbers and not make it too complicated, but I think you can handle it. So, a is 1, b is 2, and c is 3. So, a is less than b-- 1 is less than 2-- then a minus c-- so, 1 minus 3-- is less than b minus c-- so, less than 2 minus 3. All right. Let's see if that's true. 1 minus 3, that's -2. 2 minus 3, that's -1. And -2 is smaller than -1. That's what I wasn't sure if you remembered, but I'll jog your memory just in case. Think back to the number line. Let's throw up a quick number line.

(Describer) She draws a line with arrows at the ends.

We don't need much on it. Let's just get 0, 1, 2, 3, -1, -2. So, remember-- Let me clean that up so you're clear that that is a 2 there. It's easy to tell values that are larger or smaller when we're on the positive side of the number line. Intuitively, we know that 3 is bigger than 2. Ten--that would be way over here-- is bigger than 1. We know that on the positive side. Remember the negative side? The further away you get from 0, the smaller your value actually is. In this case, -2 is smaller than -1 because it's further away from 0 on the negative side of my number line. I know that's tricky, because you're like, "But 2 is bigger than 1." But -2 is smaller than -1. Because on the negative side, the further away you are from 0, the smaller your value actually is. So, this subtraction of inequality holds true. All right, let's keep going. Same thing's true for greater than. So, if a is greater than b, a minus c will also be greater than b minus c. Okay. Multiplication. Things start getting interesting here. Not at first, but they will. If a is less than b and c is positive, then a times c will be less then b times c. Let me show you what I mean. Let's say a is 1, b is 2, c is 3. They've been doing good so far. So, a is less than b, 1 is less than 2. C is positive. Yes, c is a positive number. So, a times c, 1 times 3, will be less then b times c, or 2 times 3, okay? So, 1 times 3, that's 3, 2 times 3, that's 6. Okay, that's true. If a is less than b and c is positive, the product of a and c will be less than the product of b and c. Let's keep going. The same thing holds true for greater than. If a is greater than b and c is positive, the product of a and c will be greater than the product of b and c. Here's where it gets interesting. You may remember this from pre-algebra also. Things change if c is negative. If a is less then b and c is negative, then ac is actually greater than bc. Let me show you what I mean. I'm gonna switch the values around. I'm gonna let a be 3 and b be 2, and I'm gonna let C be -1 to keep things simple. Oop, that's not gonna work, 'cause a's not smaller than b. So, let's let... a be 2 and c be 3. There we go. So, a is smaller than b, a is less than b. It is. Two is less than 3, and c is negative. It is, because we're letting c be -1. I'm gonna not write the inequality sign till the very end to show you what I mean. The product of a and c, that would be 2 times -1. That's -2. The product of b and c, that would be 3 times -1. That's -3. You see that the product of a and c is greater than the product of b and c. That always happens if your c is negative. If you multiply an inequality, if you multiply both sides by a negative number, essentially your inequality sign flips the other way. Here we had a less than b and c was negative. So, the product of a and c is greater than the product of b and c. As a shortcut way to think about it, if you're multiplying both sides by a negative number, you flip your inequality sign for your answer. Okay? Keep going. Same thing's true for greater than. If a is greater than b and c is negative, then the product of a and c will be less than the product of b and c. We started out with greater than, but in our answer, inequality switched to less than because we were multiplying by a negative number. So, division property of inequality. Same idea for multiplication. Okay. If c is positive, things are kind of normal. If a is less than b and c is positive, then a divided by c will be less than b divided by c. I think you're getting the idea of how that works. I won't bother doing a numeric example for that one. Same thing's true for greater than. If a is greater than b and c is positive, the quotient, a divided by c, will be greater than b divided by c. Same thing happens here like it did with the multiplication property. If a is less then b and c is negative, that inequality sign will switch the other way. So, a divided by c will be greater than b divided by c. And that happens if that c is negative. Same idea for the greater than case. If a is greater than b and c is negative, then a divided by c will be less than b divided by c. Let's look at an example and see if you can spot these properties while you're solving an inequality. This will throw you back to pre-algebra, when you were solving multi-step linear inequalities. We're gonna zone in where we see property symbols, or property boxes, and figure out what property allowed us to move between those steps when we get down there. Let's interpret this. Step one, just giving us the problem there. Five times that quantity x plus 2 is greater than 30. Step two looks like they applied the distributive property, right? 'Cause now we have 5x plus 10 is greater than 30. Step three, looks like they are subtracting 10 from both sides. Right? They've got 5x plus 10 minus 10 is greater than 30 minus 10. In step four, 5x is greater than 20. Okay. And we see the little property question. Which property of inequality tells us that if we subtract the same quantity from both sides of our inequality, that inequality will still hold true-- whatever that relationship was? Remember? It was that subtraction property of inequality. Okay? All right, let's keep moving. From step four to step five, let's scroll a bit. We divided both sides by 5, and from step five to step six, we went from 5x divided by 5 is greater than 20 divided by 5 to x is greater than 4. We divided both sides by 5, and the relationship between both sides of our inequality still held true. Which property of inequality told us that-- that if we divide both sides by a positive number, the relationship still holds true, whatever it was originally? That was the division property of inequality. Okay? All right. Good job on that. Take a look at this one. I'm gonna give you a bit. I want you to focus on steps one through four. Do you know what property allowed us to move between step three and step four? Then I'll scroll down, because there's one more property question lower in this problem. Hit pause, give that some thought, and press play when you're ready to go over it.

(Describer) Titles: Step 1: Negative x minus 4 is less than negative 2 times x plus 3. Step 2: Negative x minus 4 is less than negative 2x minus 6. Step 3: Negative x minus 4 plus 2x is less than negative 2x plus 2x minus 6. By this step is box with the word "property" with a question mark. Step 4: x plus 4 is less than negative 6.

(female describer) Step 1: -x minus 4 is less than -2 times x plus 3. Step 2: -x minus 4 is less than -2x minus 6. Step 3: -x minus 4 plus 2x is less than -2x plus 2x minus 6. By this step is the box with the word "Property" with a question mark. Step 4: x plus 4 is less than -6. As you made your way through here and got to step three and step four, and you saw that you had 2x added to both sides, and that initial relationship between the left side and right side of our inequality still held true. The addition property of inequality allowed that to happen. Good job if you got that. If not, review those properties and you'll get good at it. I'm gonna scroll down. Now focus on step four through step six and see if you can figure out what property allowed you to move between steps five and six. Press pause, give that some thought, then press play when you're ready to check.

(Describer) Titles: Step 5: x plus 4 minus 4 is less than negative 6 minus 4. By this step is a box with the word "property" and a question mark. Step 6: x is less than negative 10.

(describer) Step 5: x plus 4 minus 4 is less than -6 minus 4. By this step is a box with the word "Property" and a question mark. Step 6: x is less than -10. Hope you gave that some thought. If you looked at what was going on between steps five and six, you subtracted 4 from both sides and the initial inequality relationship still held true: x was less than -10. That was the subtraction property of inequality that allowed you to move between those two steps. Great job getting familiar with the properties of inequality. I hope to see you back soon for more Algebra I. Bye.

(Describer) Accessibility provided by the U.S. Department of Education.

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In this program, students learn about the properties of inequalities. For example, if the same amount is added to both sides of an inequality, then the inequality is still true. Part of the "Welcome to Algebra I" series.

Media Details

Runtime: 18 minutes

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Episode 1
31 minutes
Grade Level: 7 - 12
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Episode 2
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