It's not just numbers and variables that get distributed. Positive and negative signs can, too. In this lesson, we'll learn how to distribute positive and negative signs correctly.
The distributive property tells us that a(b + c) = ab + ac. Imagine you're b and you're in a car with your friend c. It's winter and really cold outside. The cold is a. A wants in. It wants to distribute itself throughout the car. And what happens if a door or window opens? A is distributed, and now you're ab and your friend is ac. The cold a has latched itself onto both of you.
So, that's how the distributive property works, at least in cold winter terms. But just as the cold can permeate a previously warm car, other things can distribute, or permeate. In this lesson, we'll learn about two of these distributing forces: positive signs and negative signs.
A positive sign is like happiness. It's totally, well, positive. Look at this expression: +(x + 1). That positive sign is like a rainbow just itching to get inside. And what does it do if we distribute it? We get x + 1. So, a positive sign makes no difference on the signs.
Really? A rainbow can't turn a frown upside down? It's true. Look at this one: +(2x - 3). That three is feeling kind of down, as we know from the negative sign in front of him. What if we distribute the positive sign? We can imagine the sign is a +1. What's +1 * 2x? 2x. No change there. What about +1 * -3? Still -3. That -3 is really in a funk. So if we simplify +(2x - 3), we get 2x - 3.
Let's try a couple of practice problems involving positive signs. Here's one: +(3x + 5). We have a positive sign, like bacon. Bacon is awesome! Well, if we try +1 * 3x, we get 3x. And +1 * +5? That's +5. So, our simplified expression is 3x + 5. Sorry, bacon, your charms do nothing here. Maybe our 3x and +5 are vegetarians.
Let's do one more: +(-4 - 9x - 2y). Okay, if -4 and -9x and -2y are in a room together, that's an overwhelmingly negative room. It's like a group of friends watching their favorite football team lose 43 to 8. But then there's that positive sign that wants to get distributed. It's like an adorable kitten at the door. How can that not affect you?
Well, +1 * -4 is, yep, -4. +1 * -9x is -9x. I think you see where this is going. +1 * -2y? That's -2y. So, we can distribute all the adorable kittens we want, we still get -4 - 9x - 2y, with no signs changing. Maybe they're allergic to kittens. Or maybe their football team got beaten by a team inexplicably called the Kittens, so the kitten is a reminder of that pain.
Rainbows, bacon, kittens - positive signs having no affect when we're distributing. What about negative signs? Here's an expression: -(3 + x). To simplify, or distribute the negative sign, we can follow one of two strategies. Our first option is to swap each sign. So +3 becomes -3. And +x becomes -x. So, we get -3 - x. Oh, man, it's like a flu virus that just attacked everyone and reversed all the happiness.
So, that's option one; what about option two? We could insert a 1 and multiply. Just as we did with the positive sign, we can add a 1 to the negative sign and then we have -1 * 3, which is -3, and -1 * x, which is -x. We again end up with -3 - x.
Let's practice. Here's an expression to simplify: -(8 + 3x). Okay, that negative sign is lurking outside the door like an empty box of doughnuts. Is there anything sadder than a box that was once full of chocolate-frosted happiness but is now empty? Probably, but empty doughnut boxes are sad, too. Let's try multiplying each term by -1. -1 * 8 is -8. -1 * + 3x is -3x. So, we get -8 - 3x. It's like it rained all over our expression.
Here's another one: -(-2x + 7 - 4y). This is a longer expression, so let's be careful. We still have that negative sign lurking. This one is like a rain cloud. Let's try swapping all the signs. -2x becomes +2x. Hmm, so maybe -2x likes the rain. +7 becomes -7. No love for a rainy day there. And -4y becomes +4y. Another rain-lover. So, we have positive 2x - 7 + 4y. That's our simplified expression.
To summarize, we learned about distributing positive and negative signs. The distributive property can be defined as a(b + c) = ab + ac. When we have positive signs to distribute, we can treat it like a +1, which doesn't affect any of the signs.
When we have a negative sign to distribute, we can follow one of two paths. First, we can swap all the signs. Positive signs become negative and negative signs become positive. Or, we can multiply each term by -1. That will have the same outcome.
After watching this lesson, you should be able to apply the distributive property correctly to distribute positive and negative signs of an expression.