# Practical Algebra 1.3

November 7, 2019

Section 1.3: multiplying and dividing

integers. This video will help you

multiply and divide two signed numbers, and multiply and divide several signed

numbers. When multiplying or

dividing, we use two real simple rules to figure out the sign

of the product, which is multiplication or the quotient, which is division. Opposite signs are going to give us a

negative answer, so notice if I take a positive number times a negative number

those signs are opposite. I’m gonna get a

negative answer. If we reverse that

same thing, negative times positive, I’m going to get a negative answer. The other

rule is if we have the same signs, we’re going to get a positive answer. This works whether we’re multiplying or

dividing. We’ve done this one several

times: a positive times a positive. We

know we get a positive answer. Well,

if we take a negative times a negative, we’re also going to get a positive

answer. The same thing holds true for

division. Two positives or two

negatives are going to give us a positive answer. Now, let’s take a look at

some examples that will help you understand how to multiply and divide

positive and negative integers. Example one, we’re asked to multiply

positive six times negative five. Well,

those are different signs, so we know we’re gonna get a negative answer. Once we get our negative sign down there,

we’ll just focus on whatever six times five is. Well, six times five is

thirty, so my answer there is going to be

negative 30. Example two, I’m dividing

negative 16 by negative two. Same signs

here, so we’re gonna get a positive answer. I can write that positive sign,

or I could just leave it blank. Either

way, it indicates it’s positive. Then, I

can just focus on the division. 16

divided by 2 is 8. Question three,

we’re asked to multiply negative 4 times negative 8. Again, same signs, so that’s

gonna give me a positive answer. 4 times

8 is going to be 32, so positive 32. Here,

we want to be a little bit careful. 6

divided by zero. Well, division by zero

is impossible. Basically what

happens when we’re when we’re dividing something, we’re

leading it up and we really can’t split something up into zero parts. Zero parts

really doesn’t exist, so this problem is impossible, but in mathematics we’ve

learned never to say never. What we do

when we think something is impossible is we say ‘undefined’. This means no one has

figured this out yet, so we don’t know what the answer is. Most likely, this is

gonna end up being impossible, but at this point we’re just saying, hey nobody

knows. Question five, four divided into 0. Now,

four can go into zero, and it goes in there zero times. If I take four

times zero, it gives me zero. That means 4

goes into 0, zero times. So, if I am

dividing zero by another number, I can actually get an answer there, because all

we’re doing is splitting up zero in two parts. There would be zero in each of

those parts, or you could think about the multiplication problem. Four times zero

will give you zero, so this actually works when we’re dividing zero in two

parts. We get zero, but if we’re trying to

divide by zero, then we end up getting undefined. Pause your video player

now, and answer these practice questions. When you’re done, hit play to see how you

did. Question six, we’re asked to

multiply two negatives. We know

that’s going to give us a positive answer, then we can focus on seven times

eight. It is gonna give us 56, so

positive 56. Question seven,

different signs here, so we’re gonna get a negative answer. Then, we would say

ten divided by five is two, so negative two for our answer there. Question eight has different signs so it is

a negative answer. Nine goes into 36

four times, so negative four. Question

nine, we’re starting with zero and we are dividing that into two parts. We can do

this. We can split zero up into two parts. Each of those parts would have zero. Question ten. Now, we’re asked to figure

out how many times zero goes into twelve. Well, it’s not gonna work because we’re

really essentially dividing by zero. This is going to be undefined. Multiplying and dividing several integers. Two simple rules can be used to

determine the sign of several products. Again, that’s multiplication or quotients

which is division, or a combination of the two. The first rule is this: if we

have an even number of negatives in our list of things getting multiplied

together, we’re going to end up being having a positive answer. You can see

here in this first list, I have two negatives, which is an even number of

negatives. We’re going to end up with a positive

answer here, the reason being that every time I pair up two of these negatives, I’m

gonna get a positive answer. When I

pair those two up, it’s gonna give me a positive answer, so if there’s an

even number, they all get paired up. That’s how we end up with a positive

answer. Second example,

I have one, two, three, four, number of negatives there, so that’s also going

to give me a positive answer. The next rule

is this: an odd number of negatives gives us a negative answer. If we’re

looking at this problem right here, I have one negative, so that is gonna give

me a negative answer. The second

one here, I have one, two, three, negatives, so that’s also going to give me a

negative answer. The way this works is

basically these first two negatives would pair up to make a positive, and

then we have one negative left over. That one negative left over is

going to give us a negative answer because our signs will be different, a

positive times a negative. Let’s take a

look at an example that will help you understand this. Example 11,

we’re asked to evaluate, which means get this down to a single answer. I’m

gonna multiply and divide here. What

we do with multiplication and division is we work left to right because those

tie in the order of operations. I would start here, and multiply the

negative 3 times a negative 5. That’s gonna give us a positive 15. I

rewrite the rest of the problem still working left to right. Now, 15 times

2 is going to give me 30. Come over here

to finish, and then the last step 30 divided by 6, is going to give me a

positive 5. Now, let’s double check that we’ve done this

work. Let’s double check and make sure

the sign on this answer, is what it should be. If I go back to the

original problem, I had an even number of negatives in there, so that should give

me a positive answer. I ended up with

a positive answer, so I have confidence in that answer. Go ahead now, and pause

your video player and answer practice question 12. When you’re done, hit play to

see how you did. Question 12, some multiplication and

division. here We’re going to work

left to right, so I’d start right here 6 and negative 2. That’s gonna give me a

negative answer. 6 divided by 2 is 3.Then,

I’m gonna rewrite the rest of the problem. Still working left to right,

negative 3 times negative 8. The signs are

the same, so that’s going to give me a positive answer. Positive 24 divided

by negative 4. It’s

going to give me a negative answer, and then 24 divided by 4 is gonna give me 6. I get a final answer there of

negative 6. Go ahead and pause your video

player again, and work some of these practice questions. Maybe work these 13

through 18. When you get done, hit

play, and I’ll show you the answers. Take a look

at the answers here, and see how you did. Let me talk a little bit about 16

because 16 was a little tricky. Negative 4, while it does divide into

positive 10, it doesn’t go in there evenly. I think it might be a little

easier to think of this, instead of writing the division this way in long

division form. Writing it over here as a

fraction is something we’re probably more used to. I know my answer is gonna be

negative because I have a positive and a negative so I wrote the negative sign. Then, 4 goes into 10 two whole

times with a remainder of 2. Then you

write that remainder over the original divisor. So, really what we end up with

here is negative two and two fourths, and then when we simplify two fourths, we get

1/2. Our final simplified answer there

should be negative two and a half. Now, it’s time to go ahead and pause

your video player again, and let’s get some more practice. I’ve mixed it up

here, so we have both addition, subtraction, multiplication, and division. That’s one of the harder things to

do is to kind of keep the signs all straight, and then the rules all straight

as we do this. Go ahead, and work these

practice problems 19 through 32. When you’re done, hit play and you can

check your answers. I’ll post some

answers here for you. Go ahead, and

check your answers now and see if your answers match up with the answers I have. If you’ve made a mistake, go back

and try to correct it. It’s easy to make

mistakes here and get your own signs and your operations mixed up, so go ahead and

double check these and see how you did. If you made a mistake, go ahead and fix

it. Go ahead, and pause your video

player again, and now work questions 33 through 44. When you’re done with those, go ahead and

hit hit play. I’ll post some answers here for you. Go ahead, and check your answers now, and

see if you came up with the answers I have here. If you got something different,

go back and try to rework it, and see if you can get the correct

answer. Last two problems. Go ahead, and

pause your video player and work 45 and 46. See if you come up with

the same thing I’ll show you here in just a second. Go ahead, and check

your work here, and see if it matches up with mine. If your steps follow the

same steps that I have here, if you made a mistake go back and try to fix it.

On problem 36 you multiplied (-4) and (-8) even though there's a division sign. Shouldn't the answer be .5 or 1/2?