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The Greater Than Sign
The greater than sign is an arrow symbol that points to the right. It is most commonly used to compare numbers, write inequalities, or translate English sentences to math sentences.
English Sentence
"8 is greater than 2"
Math Sentence
\(8>2\)
How to Use the Greater Than Sign
To use the greater than sign...
- Write the largest number first.
- Write the greater than sign (>)
- Write the smaller number last.
Example: Compare 4 and 7
Step 1
7 is the largest number
Step 2
>
Step 3
4 is the smallest number
Answer: \(7>4\)
Less Than Symbol vs Greater Than Symbol
The less than symbol (\(<\)) and the greater than symbol (\(>\)), can both be used to communicate the same information.
For example...
"6 is greater than 4"
"4 is less than 6"
\(6>4\)
\(4<6\)
Both of these sentences tell us that the 4 is the smaller number and 6 is the bigger number. But the symbols are different depending on which number is listed first.
Because \(<\) and \(>\) are so similar, I like to use the alligator trick or the arrow trick to help me remember which symbol I should use.
Alligator Trick
If you imagine that the greater than symbol and the less than symbol are an alligator's mouth, the alligator will always eat the "most desirable number".
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Arrow Trick
The smaller side of the arrow is next to the smaller number.
The larger side of the arrow is next to the larger number.
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Greater Than or Equal To Symbol
The greater than or equal to symbol (\(\geq\)) is a combination of the greater than sign (\(>\)) and the equal sign (\(=\)).
It is used to write inequalities where the first number EITHER has to be greater than the second number OR both numbers could equal each other.
Example: List possible solutions for \(x\geq y\)
\(x=3\)
\(y=1\)
3 is greater than 1 so this is a solution
\(3\geq 1\)
|
\(x=2\) \(y=2\) |
2 is equal to 2 so this is a solution \(2\geq 2\) |
\(x=4\)
\(y=5\)
4 is less than 5 so this is NOT a solution
\(4\ngeq 5\)
When you are graphing one-variable inequalities, the greater than symbol is represented with a closed circle while the greater than or equal to symbol is represented with an open circle.
Similarly, when you are graphing two-variable inequalities, the greater than symbol is represented with solid line while the greater than or equal to symbol is represented with a dotted line.
Comparing Negatives and Positives
When you are asked to compare numbers, it can be helpful to put them on a number line, especially if any of the numbers are negative.
The number that is farthest to the right on the number line is the largest number.
That means that all of these statements are true...
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4 > 1 4 > -2 |
1 > -2 1 > -4 |
4 > -4 -2 > -4 |
The reason 1 is greater than -4 is because 1 is a positive number. A positive number will always be greater than a negative number.
That is why 4 is greater than -4 instead of being equal.
Both -2 and -4 are negative numbers but -2 is closer to the positive number side of the number line so -2 is greater than -4.
Debt Analogy
There is a debt analogy that I like to use when I'm comparing positive and negative numbers because it makes it super easy to tell which number is greater than the other.
In this analogy, positive numbers represent an amount of money you are given as a gift. Negative numbers represent an amount of money you are in debt. The "better" number is greater than the "worst number".
In this analogy, the reason 1 is greater than -4 is because it is better to have $1 given to you as a gift than to be in debt -$4.
The reason -2 is greater than -4 is because it is better to be in debt -$2 than to be in debt -$4.
"Bigger Than" vs Greater Than
Sometimes it is useful to partially ignore the negatives when you are comparing numbers.
For example, 3 is greater than -6 because -6 is a negative number and 3 is a positive number. But when you add or subtract negative numbers, it is useful to be able to say that -6 is a "bigger" number than 3.
The mathematically accurate way to say this is, "The absolute value of -6 is greater than the absolute value of 3."
Or in math language...
\(|-6|>|3|\)
However, I think it feels very clunky to have to say, "The absolute value of ___ is greater than the absolute value of ___" every time.
So, I like to use the phrase "bigger than" whenever I want to ignore the negatives and compare the magnitude of the numbers.
It is important to realize that "bigger than" is very different from "greater than" because bigger than ignores the negatives while greater than takes them into consideration.
Greater Than
-2 is greater than -9
\(-2>-9\)
"Bigger Than"
-9 is "bigger than" -2
\(|-9|>|-2|\)
It is also important to realize that "bigger than" is NOT a standard mathematical term. I'm also pretty sure that I am the only math teacher that uses this phrase.
So, if you decide to use this phrase around your math teacher, make sure to tell her (or him) that when you say...
"___ is bigger than ___"
That means...
"The absolute value of _ is greater than the absolute value of _."
If you don't clarify this for your teacher, she (or he) will probably assume that "bigger than" is the same thing as "greater than" and your answers may be marked wrong.
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