Let’s take a look at comparing and ordering integers. Here’s the key to ordering and comparing integers. When two numbers are graphed on a number line, the number to the left is always less than the number on the right. The number to the right is always greater than the number to the left. Here negative two is to the left of zero. We would write it in this way, negative two is less than zero. Here’s another case. The one is to the right of negative three. We would write one is greater than negative three because the one is to the right. Let’s check out some examples like this one.
Negative 12 is less than negative 13, true or false?
Let’s use a number line to think through this. Here’s what it looks like when we count with our negative integers. So I extended out our number line and added negative twelve and negative thirteen in their respective places. So the rule is, the value to the right of the other value on the number line is larger. In other words, as we move to the right on a number line, the values are getting bigger. In this case, negative 12 falls to the right of negative 13, which makes negative 12 a larger value than negative 13. The problem in our example lists negative 12 as less than negative 13, which makes this statement false. Negative 12 is not less than negative 13. Here’s another example, example two.
Fill in the box with greater than, less than, or equal to. Negative 96 is greater than, less than, or equal to the absolute value of negative 96.
Remember, absolute value is a numbers distance from zero on a number line. Before we move forward in answering this question, we need to figure out what the absolute value of negative 96 is. The distance from negative 96 to zero on a number line is 96 units. Now we’re looking at something like this negative 96 compared to 96.
Back to our number line again, as we moved to the right of the number line, our values are increasing, so we can say that negative 96 is less than 96. Or in the original example, negative 96 is less than the absolute value of negative 96. We don’t just want to be able to compare two different values though. Sometimes we want to order values. When we’re ordering values, we’re taking a whole set and we’re putting them in order, from least to greatest, or from greatest to least.
Example three is asking us, order the following from least to greatest. Negative seven, nine, 18, 14, negative 11, negative 10.
Back to our trusty number line, we’ll put our zero in the middle. Okay. Now I’ve just divided up the values that we’re working with and I put the negatives on the left side of zero, because I know that’s where they’re going. And I put all of the positive values on the right side of zero, because I know that’s where those will go. Ordering the positive values will not be difficult. Nine is the smallest of the three, next comes 14, and then 18. Those are good to go. But we have to be a little bit more careful when we’re working with negative values. We can actually use absolute value to help us do this. We know that the absolute value of a number is its distance from zero.
So as I’m placing negative seven, negative 11, and negative 10 on the number line, first I’m gonna find their absolute value. Negative seven would fall seven units away from zero on a number line. Negative 10 has an absolute value of 10, so it’s the next number in our list. It falls 10 units from zero on the number line. And finally, our negative 11 falls 11 units from zero on the number line. But our work here is not done. I said a few times as we move from left to right on a number line, our values are getting larger.
This is exactly how we number least to greatest. We start at the furthest left and we move to the furthest right. That’s what the list would look like, ordered from least to greatest. But what if we change the problem, and we try to order them from greatest to least. The opposite would be true here. If we start to the far right and move to the left, the values are getting smaller. And when you want to move from the greatest to the least, you would then wanna look at the right and then move to the left. This is what an ordered list moving from greatest to least would look like, 18, 14, nine, negative seven, negative 10, and then negative 14 being the smallest.
Here’s our last example. A helicopter is hovering at a height of 647 feet, a diving bell is at negative 567 feet. Which one is farther from sea level?
Our question, which one is farther from sea level, the two pieces of information that were given. But we also have to remember, what would it mean to be at sea level. What number value is sea level? If something is directly at sea level, its zero feet. In our case, let’s put our helicopter here at 647 feet, and we put the diving bell here.
Our question is asking which one is farther from sea level. Another way we could say this is, which one is further from zero. The diving bell is 567 feet below sea level, or 567 feet from sea level. And our helicopter is 647 feet from sea level, or from zero. 647 is greater than 567. Did you catch that we’re actually using absolute value here? We found the absolute value of negative 567. The final answer here would be the helicopter. The helicopter is farther away from sea level. Good luck! And see you next time.