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Positive and Negative Numbers - I


Mike McGarry
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 aims to build proficiency in these fundamental mathematical skills, crucial for SAT exam preparation.
  • Introduction to basic arithmetic operations with positive and negative numbers, emphasizing the importance of mental math skills for the SAT.
  • Explanation of the concept that subtraction can be rewritten 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 exercises and examples to reinforce understanding and application of these arithmetic concepts in various scenarios.
  • Highlight on the universality of these techniques, applicable to integers, decimals, and fractions, and their significance in building a strong mathematical foundation.
Chapters
00:00
Understanding Basic Arithmetic Operations
01:07
Subtraction as Addition of Opposite Sign
03:01
Simplifying Double Negatives and Reversal of Subtraction
03:37
Practical Exercises for Mastery

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