Amy has a master's degree in secondary education and has been teaching math for over 9 years. Amy has worked with students at all levels from those with special needs to those that are gifted.
Picture two pie slices with one being bigger than another. We can use inequalities to tell us which is bigger. What exactly is an inequality? Think of it as comparing the sizes of two values. The word 'inequality' literally means not exactly equal. So when we are using inequalities, our two values may not be equal and it is our job to figure out which value is the larger one.
To help us do this, we have symbols we can use. And these symbols actually are very good at telling us which one is bigger. For example, to say that 2 is greater than 1, we use the greater than symbol ( > ) and write 2 > 1. You can remember this sign easily because it looks like a mouth that is opening towards the bigger number. Who wouldn't want to eat the bigger number? It's like getting the bigger slice of pie!
The Four Inequalities
But, did you know that we have three other inequality symbols at our disposal? If we flipped our sign to look like this ( < ), we now have what is called the less than symbol. That's right, you guessed it! The smaller number is written first and the larger number second so that the mouth is towards the larger number. So we can write, for example, 1 < 2, and we read it as 1 is less than 2.
The other two symbols involve both the greater than and less than symbols. The only difference is that it tells us that we can also have the two sides equal. So we have the greater than or equal to symbol ( >= ) and the less than or equal to symbol ( <= ).
Greater than or equal to:
Less than or equal to:
For example, we can write 6 <= 6, or we can write 5 <= 6. We can also write 6 >= 6 or 7 >= 6. I can write 6 <= 6 or 6 >= 6 because both symbols allow for the numbers to be equal.
True or False Statements
Here is where we get into true or false statements. Sometimes we are given a problem that asks whether a particular inequality is true or not. For example, we may be asked to determine whether the statement 80 > 90 is true or not. How do you work with these problems? Well, you use your knowledge of counting and which numbers are larger to compare the numbers, and then you look at the symbol to see if it is the correct one.
So, to determine whether the statement 80 > 90 is correct, we first compare the two numbers, 80 and 90. Which of these numbers is larger? 90 is, of course, larger than 80. Now we look at the symbol used. We see the symbol used is the greater than symbol. So, we read, 80 is greater than 90. But wait, that's not true because you know that 80 is smaller. So this means that the statement is false. If we rewrite the statement as 80 < 90, we would then have a true statement.
When we add a variable into the mix, as we are bound to do since this is algebra, we end up with what we call an open sentence, a statement that includes a variable. We call it open because we don't know for certain whether a statement is true or false. For example, the statement x > 2 is an open sentence because the statement can be either true or false depending on what x is.
Let's review what we've learned, now. We learned that an inequality compares the sizes of two values. We have four different ways to compare values. We can have one value greater than ( > ) another; one value less than ( < ) another; one value greater than or equal to ( >= ) another; or we can have one value less than or equal to ( <= ) another. If the inequality is true from what we know about numbers, then we call the statement a true statement. If it is false, then we call it a false statement. If there are variables involved, then we call it an open sentence.
Once you have completed this lesson you should be able to:
- State the 4 possible inequalities in math
- Determine if an inequality is true, false or an open statement
- Compose a true, false, or open sentence inequality
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