Absolute value

The most important thing I want you to remember after reading this is that absolute value is a measurable distance.

The second most important thing I want you to remember is that absolute value is positive most of the time. (This is because distance is generally a positive measurement.

Absolute value is indicated by two long straight lines |  |. This means you’re trying to find how far you are from 0. For example, |4| is four units from 0, and it therefore it equals 4. If you’re dealing with |-4|, that’s also 4 units away from 0, so it also equals 4.

That’s it…that’s all there is to absolute value. Of course, then you have to learn how to apply it.

Let’s say you’re given the problem -| -6|. You know the absolute value of -6 is 6, but because there’s a negative sign outside the absolute value signs, it stays with the 6 for an answer of -6.

Equations sometimes have absolute value in them. Look at the following:  |2x – 4| = 8  Because the equation inside the absolute value signs can be positive or negative, we have to solve for both answers.

2x – 4 = 8                                       2x – 4 = -8

2x = 12                                           2x = -4

  x = 6                                               x = -2

The solution to |2x – 4| = 8 is both 6 and -2.

Absolute value can also show up in inequalities. Let’s look at |3x + 9| > 6  Again, we have to solve two equations, and we have to pay very close attention to the set-up.

3x + 9 > 6                                      3x +9 < -6

   3x > -3                                           3x < -15

   x > -1                                              x < -5

Note that when we set the second equation to the negative answer, we also flipped the inequality sign. Our solution is x > -1 or x < -5. If we were to graph this out, we would have a broken line with a gap between -1 and -5.

If you’re not sure you did the inequality correctly, you can always pick a number that would make the sentence true and check your work. If it doesn’t make a true statement once you simplify the problem, you probably have the inequality sign going the wrong way.

That’s it for this week. If you have any suggestions for next week, or questions about work you’re doing in school, feel free to drop us a line at deadbunnyed@gmail.com.

Multiplying and dividing integers

Multiplying and dividing integers is actually an easier process than adding and subtracting them. You should already know your basic facts. That’s the first step.

The second is to memorize these two rules.

• If you multiply/divide numbers with the same sign, the answer will be positive.
•  If you multiply/divide numbers with different signs, the answer will be negative.

So, in practice. the first rule looks like +3 * +5 = +15, or -4 * -7 = +28, or +12 / +4 = +3, or -8 / -4 = +2. The second rule looks like +4 * -4 = -16, or -42 / +6 = -7. It doesn’t matter which number is negative in multiplication or division. If one is negative and the other is positive, the answer has to be negative.

So we’ve covered the rules for working with integers. Someone suggested I do a little bit on absolute value, so I’ll try to post on that tomorrow.

Subtracting integers

Yesterday, we covered adding positive and negative integers. Today, let’s look at subtracting positive and negative numbers.

First, let’s remember the basic rules of adding and subtracting.

Subtraction always goes left on the number line. Addition always goes to the right.

All right, let’s start with something simple: 7-4. This problem is actually +7 – +4, and is just like the subtraction problems you’re used to doing. You start at 7 on the number line and hop down 4 spaces, landing on +3. So, +7-+4 = +3

This is all good, right? A positive minus a positive is a positive, right? Well, not exactly.

Let’s look at 4 – 7. This is now +4 – +7. We start at +4 on the number line, hop down 7 spaces (which takes us across the 0), and land on -3. When subtracting two positive numbers, the sign on the answer will match the sign on the larger number in the problem. Above, it was positive because the 7 was positive. This time it was negative because of the minus sign in front of the 7.

Now let’s look at what happens when both numbers are negative. Let’s try -2 – (-6). If you remember, yesterday I said we don’t like having two symbols in between numbers, so we combine them. Because these two signs are the same, they combine and become a plus sign. This problem is now -2 + 6. We start at -2, hop 6 spaces to the right on the number line, and arrive at +4.

Subtracting negatives is like subtracting positives because the sign on the larger number, when the expression is simplified, is the same as the sign on the answer.

Tomorrow, we’ll tackle multiplying integers.

Adding integers

Hi, everyone! I hope you all had a great summer! School is starting back up here next week, so I figured now was as good a time as any to start the blog back up.

This week, we’ll be looking at working with integers. Let’s start with a definition. What is an integer? An integer is a number with no fractional parts. Most people would say, “Oh, it’s a whole number,” but that’s only half true. Whole numbers are only positive. Integers include negative numbers and zero.

I want you to remember one other thing before we get started.

On a number line, addition always goes to the right and subtraction always goes to the left. This is important in understanding how to work with integers.

Let’s look at a familiar problem: 5 + 3. This problem is actually +5 + +3. We’re adding two positive integers. We start at 5 on the number line and move three spaces to the right, landing on +8.

But we already know 5 +3 = 8, and we know that if we add two positive numbers, we’ll end up with a positive number.

Now let’s consider a less familiar problem: 5 + (-3). We’re now adding a positive number and a negative number. We don’t like having two operations in the middle of the equation, though, so we simplify. Because the signs are different, the negative sign wins, turning into a minus sign.

The problem now turns into the more familiar problem 5-3. We start at 5 on the number line, and move three spaces to the left to land on +2.

Because the positive number was the larger number, the answer is positive. Had the larger number been the negative number, then the answer would have been negative. For example: 3 + (-5) = 3 – 5 = -2.

Sometimes, we’re asked to add two negative numbers. Let’s look at -4 + (-2).

We know from our earlier work that this problem is really the subtraction problem -4 – 2. We’ll start at -4 and move two spaces to the left to land on -6. -4 – 2 = -6.

Because we’re adding negative numbers, the answer will be negative.

You’ve probably noticed that in two cases, the addition problem actually became a subtraction problem. What happens in subtraction, then? We’ll look at that tomorrow!