4.1 SAT Math - Negatives

Most of the more advanced number concepts we'll review here for the SAT math section are things that have been touched on in the Algebra Basics review, or may not seem that advanced, but they will be tested and could probably use a quick review before getting into the more complicated versions.

Negatives

Hopefully by now you know what negatives are and how to work with them, but just remember that on a number line, negatives are to the left of zero:

The general rule for negatives is "one of each, think minus sign; two of the same, think plus sign."  What does that mean?

Well, first things first.  No number should have more than one sign.  Everything except 0 is either positive or negative, right?  So if you're given more than one sign, you should always clean it up first. 

-(-3) + 4 - 8 + (-2)

Get rid of any double signs.  If there are two of the same sign, it becomes positive.  If there's one of each, it becomes negative. So now you're left with:

3 + 4 - 8 - 2

Now we can add/subtract.  Some teachers teach this differently, but usually the easiest way to add/subtract it to remember that same rule:  "One of each, minus sign; two of the same, plus sign." 

So going left to right we have positive three and positive four.  That's two of the same sign, so we add and get 7.  The answer is obviously positive because 3 and 4 were both positive.  Next.

7 - 8

What do we have?  One of each (a positive 7 and a negative 8).  So?  Minus sign.  That means, forget about the order they're in or who has what sign and just subtract them like regular numbers, big minus small.  8 - 7 = 1  Now, should that 1 be positive or negative?  Well which number (regardless of sign) is bigger, 8 or 7?  8 is a bigger number.   Since 8 was negative in our original equation, our answer is negative.  So -1.  Next.

- 1 - 2

We have two of the same again, so we think plus sign and we add.  1 + 2 = 3, but again, we have to check our original signs.  They were both the same sign, which is why we added, but they were both  negative, so our answer is negative.  So the answer?  -3.

In short, when adding or subtracting negatives:

Step 1) Look at your signs to determine whether to add or subtract.
Step 2) Ignore the signs and simply add or subtract as if they were regular numbers.
Step 3) Look to the original signs again to determine if the answer is positive or negative.

The same "One of each; two of the same" rule applies to multiplying and dividing.  Two negatives multiplied or divided together make a positive; one of each sign makes a negative.
Doesn't make sense or you learned it differently in school?  Doesn't matter.  You have a lovely little calculator which you know you should be using as often as possible anyways, right?  Right??  But that brings us to:

-Calculator Rule #2 : Know your rules; your calculator will only do what you tell it.  If you get an answer that doesn't seem right, check your work or go with your gut.  If you multiply two negatives together and get a negative, you probably forgot to use parentheses.  If you know your rules and know that what you entered into your calculator is wrong, answer the choice that you know to be correct and/or double-check what you entered to be safe. The calculator will only do what you tell it, so tell it properly or know what you did wrong.

The hardest part for most people when dealing with negatives is least and greatest.  Since negatives go left on the number line, that means smaller numbers are actually greater, since they are closer to zero.  Think of it like being in debt.  If you owe somebody $20, you technically have a net worth of -20.  If you only owe someone $5, you have a net worth of -5.  You're worth more money with the smaller number even though either way you are in debt.

That may seem a simple concept until you get into multiple decimal places.  Take this one:

Example Problem

3. Which of the following answer choices has the greatest value?
(A) -1.736
(B) -1.836
(C) -1.734
(D) -1.978
(E) -1.783

-Hint:  When comparing negatives, change the signs to positive and look for the opposite of what you were asked.  Then go place value by place value to compare.  Don't try to compare the whole number at once.

Using the above hint, we'll change all of them to positives and then look for the least.  Now, go place value to place value.  They all have a 1 in the ones place, so that's no help.  How about tenths?  7, 8, 7, 9, 7.  Which one is the least?  7.  So eliminate (B) and (D) as answer choices.  Hundredths place for (A), (C), and (E)?  3, 3, 8.  Get rid of (E).  Thousandths place?  6 vs. 4.  Four is the least so (C) is the answer.

Example Problem 2

Negatives can and will appear on number lines, but more often, number line questions will be a bit more complicated.  Take this one:

18. On the number line above, there are 8 equal intervals between 0 and 1.  What is the value of x?

For most number line questions, you'll need to count the number of spaces to determine the interval, and then do something with those numbers.  

-Hint: Number lines are often drawn to scale, which means the lines will be as they appear.  If they appear evenly spaced, they are.  Simply count the number of spaces (not the number or lines) to determine how many parts the number line is divided into.   In the above example, there are 8 spaces between 0 and 1, so each line represents

-SAT Math Tip: If a number line has the phrase "Note: Figure not drawn to scale" assume NOTHING about the spacing of the intervals.  If they give you points or digits on the line, that information is in the correct order from least to greatest, but do not assume they are close or far apart if the figure is not drawn to scale.