Positive and Negative Numbers - I

Mike McGarry
Lesson by Mike McGarry
Magoosh Expert
Frequently Asked Questions

Q: When we factor out the negative sign, do we multiply the whole equation by a negative?

A: Short answer: We are multiplying the *side of the equation that we factored the negative out of by (-1). (Not the whole equation; the examples in the video were not factoring out a (-1) from the whole equation.

We are of course not multiplying the whole side of the equation by a negative that has come from nowhere. We are taking out a negative sign from each term in the equation, and then we leave that negative sign outside of the terms as a coefficient. If we distribute the negative sign back to each of the terms, we should get our original expression. We can't forget to leave that negative that we've taken out of each term; otherwise we'll be changing the value of the equation, and we'll get the wrong answer.

We can do this with equations with numbers and with equations with variables:

  • -12 - 37 = -49

Take out a negative from each term on the left-hand side, leave the right-hand side as it is:

  • (-1) * [12 + 37] = -49

---> (-1) * 49 = -49

Check :-) Both sides of the equation are equal.  If we were solving it of course, we wouldn't know from the beginning that -12 - 37 = -49. This was to check that distributing out the negative from one side of the equation didn't change the equation.

Here's a case from the video with 2 positives, let's leave off the answer at first:

  • 62 - 74 = ?

Pull out a negative from each term on the left-hand side, the right-hand side is unaffected:

  • (-1) * [-62 + 74] = ?

---> (-1) * [12] = -12

So 62 - 74 = (-1) * [-62 + 74] = -12

And here's an example with a variable (I'm just using this as an example, it doesn't really make it easier to take out the -1 in this case):

  • -10x - 24 = 66

---> -10x = 90

---> x = 90/-10  x = -9

  • -10x - 24 = 66

--> -1(10x + 24) = 66

---> (10x + 24) = -66

---> 10x = -66 - 24

---> x = -90/10  x = -9

The mathematical reason why we have to put the (-1) as a coefficient for the terms we pulled it out of is because what we are actually doing when we pull out a (-1) from each term on one side is dividing that side of the equation by -1. In order to keep the value of that equation unchanged, we need to also multiply that side of the equation by (-1) to cancel out the division:

  • -10x - 24 = 66
  • (-1/-1) = 1

If we multiply one side of an equation by 1, that doesn't change its value. So we can do that here :-)

  • (-1) * [(-10x/-1) - (24/-1)] = 66

--->* (-1) * [(10x) - (-24)] = 66

---> (-1) * [10x + 24] = 66

Same thing as -10x - 24 = 66

Learn More About Magoosh ACT
Arithmetic and fractions, positive and negative numbers. In this video, we will discuss how to add and subtract positive and negative numbers. Now, this is a very basic topic. Again, if you are proficient in this topic. Do not feel compelled to watch this entire video.

This is a video designed to get people comfortable with this topic if it is unfamiliar or they have some uncertainty with the topic. So wherever you are starting, I will assume that you're proficient with two basic cases: how to add two positive integers or how to subtract two positive integers when they're in the form bigger minus smaller. First thing I'll say is if you need practice with this, practice every day.

It's very important to be proficient in two digit addition and subtraction. Be able to do that as mental math, that will make the test much smoother. Now the good news is if you can do these two things you can do anything else. This entire topic is very easy if you know these two things. There are many ways to discuss this material. Let's begin with subtraction.

Some mathematicians would say, Subtraction doesn't really exist. What does that mean? Well, subtraction of any number can be re-written as the addition of a number of the opposite sign. And so, some mathematicians would say that this addition is actually the true form. So, let's make sure we understand this.

Subtraction of any number can be re-written as the addition of a number of the opposite sign. Here are four different instances of subtraction. We have a positive minus a positive, a negative minus a positive, a positive minus a negative, and a negative minus a negative. In all four cases we could rewrite that subtraction as addition of a number of the opposite sign.

In the cases where we are subtracting a positive, that's the same as adding a negative. In the cases where we are subtracting a negative, that's the same as adding a positive. Well notice that we get some simplification, but it's not a simplification in every case.

For example, in the first one, it looks like we were better off where we started, we were better off without changing it to addition. In the third one, it looks like we've clearly made things better off by changing it to the addition of two positive numbers. In that fourth one, notice that now it's addition, it's commutative, so we can switch the order around, and when we switch the order around we can rewrite it as subtraction and that's much easier.

So sometimes this is really an important move for simplification, but not always. You don't always have to rewrite subtraction as addition but it can be a very good simplifying trick to have up your sleeve. Notice in particular for the, case (positive)- (negative) this trick will always simplify it. It will always become positive plus positive which is one of those fundamental things that I assume you know how to do already.

Now, let's look at that tricky double negative case, which could appear in the form negative minus positive. Or in the form negative plus negative. The big idea is we can always factor out a negative sign. Now what does this mean exactly? Let's look at that first one. (-46) -37.

We can factor out a negative sign. If we factor out a negative sign, everything inside becomes positive. So it just becomes 46 + 37, addition of two positive numbers. So you perform that addition and then just stick a negative in front of the sum. You might wanna try these others on the page. Pause the video here.

Try these others. And then you can compare your answers to mine. Here are the answers. One other case folks find tricky is the case small positive minus big positive, which also shows up as small positive plus big negative. Here the big idea is factoring out a negative sign reverses the order of subtraction.

So, what does this mean exactly? Suppose I have 23- 64, if I factor out a negative sign, then what I get is subtraction in the reverse order, 64-23. Well, now, that's bigger minus smaller, that we can do, that's one of the fundamental skills, so do that subtraction and then just stick a negative sign in front of it.

Let's look at another one. 26- 63, factor out the negative and we got a negative in front of the reversed order subtraction, 63- 26, perform the subtraction and stick a negative sign in front of it. Here are some more, you might want to pause the video here and practice these on your own.

Here are the answers I get. Very important that you're able to do things like this, and it's very good practice for mental math to be able to do this in your head. These ideas allow you to change any addition or subtraction into either the sum of two positives or the difference of larger minus smaller. Here I just discussed integers with simplicity, but all these same ideas would also be true for positive and negative decimals and fractions.

The core skills are addition of two positives or larger positive minus smaller positive. From here, if we're doing positive minus negative, we can change that to positive plus positive. If we have the double negative case, we can factor out a negative sign. And whenever we have smaller minus bigger.

We can factor out a negative sign, reverse the order subtraction. And then, what's inside bigger minus smaller, that's something we can do.

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