### Positive and Negative Numbers - I

Lesson by Mike McGarry
Magoosh Expert

Summary
The content provides a comprehensive guide on handling arithmetic operations involving positive and negative numbers, focusing on addition and subtraction. It emphasizes the importance of mastering basic arithmetic to simplify more complex mathematical problems encountered in the ACT exam.
• Introduction to adding and subtracting positive and negative numbers, highlighting the necessity for proficiency in basic arithmetic.
• Explanation of the concept that subtraction can be reinterpreted as the addition of a number with the opposite sign, offering a simplification strategy.
• Detailed walkthrough of handling double negatives and the reversal of subtraction order when factoring out a negative sign, to simplify calculations.
• Practical advice on practicing these arithmetic operations daily for mental math proficiency, crucial for smoothing the test-taking experience.
• Assurance that mastering the addition of two positives and the subtraction of a larger positive from a smaller positive equips learners to tackle any addition or subtraction problem, including those involving decimals and fractions.
Chapters
00:00
Fundamentals of Arithmetic Operations
01:07
Subtraction as Addition
03:01
Simplifying Double Negatives and Reversal in Subtraction
03:37
Practical Exercises and Mental Math

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