In this video we're going to look at translating algebraic inequalities and that means that we're going to start out with a math problem written in words and we're going to see how we can translate it into an inequality using variables and numbers and symbols. So here's a review of some common math words and how we translate them. When we have sum of or increased by that will become addition, difference of or decreased by will become subtraction product of or times becomes multiplication and quotient of or divided by become division. Now this is just a small sample of types of math words but you'll come across different ones. The important thing is to really think about what the word problem is trying to say so that you can understand how to translate it the correct way. And then we can also have unknown numbers in these problems and when we don't know what the number is, we can just represent it with a variable. Now something that's new that we're going to see in this lesson is different ways that the word is is used. Now we're used to seeing is being translated into an equal sign but as we're going to learn today, it only becomes an equal sign in certain situations. So when the word is is followed by a number, a variable or some type of operation like addition, subtraction, multiplication or division, then that word is will translate into an equal sign. However, if the word is followed by an inequality word, then it's going to translate into an inequality symbol. So here are some examples of inequality words. Now remember to focus on what inequalities represent. Inequalities represent that we're comparing two different things. So the words that we're going to come across represent comparison. For example, greater than or more than those words will translate into the greater than symbol. Less than or fewer than will translate to the less than symbol at least no less than or a minimum of will become the greater than or equal to sign. At most, no more than or maximum of will become the less than or equal to sign. So let's look at our first example. A number decreased by six is less than 14. When we see sentences like this we can go step by step and look at each part of this sentence to see how we're going to translate it into an inequality. So first we have a number and whenever we have an unknown number we're going to use a variable to represent it. You can use whatever letter you want and then we have decrease by that becomes subtraction and then we have is less than. Now remember we have those inequality words. There less than. So we have the word is but it's followed by inequality words. So that tells us that we're going to use the less than sign. So if we put this together we'll have a number but we use the variable such as x decreased by. So we have subtraction sign six is less than, we have the less than sign 14 and we can just read it as x minus six is less than 14. Here's another example. The quotient of a number and nine is at least negative four. So quotient of translates to division. So that tells us that we're going to have two things that we're dividing by each other in the next part. And what are those things? We have a number, so we'll use a variable to represent that and then we'll be dividing that number by nine. And then we have is at least we have the word is, but it's followed by inequality words. So we're going to use an inequality symbol and in this case we'll use the greater than or equal to sign. So when we put it together, we'll have the quotient of a number, which we use the variable x and nine. So we're going to show that they're being divided by each other. X divided by nine is greater than negative four. So how do we know when we're going to translate our word problem into an equation or an inequality? Because remember, the word is can be used in either situation. So let's look at some examples to help us sort this out. Ten is the difference of a number and two. Let's focus on the part of the sentence that has the word is is the difference of. Now remember difference of refers to subtraction. So we have an operation here and when is is followed by a mathematical operation, it translates into an equal sign. So we'll have an equation here, so we have ten equals the difference of a number. So we use a variable like x and two. So x minus two. Next we have ten is five more than a number. So if we focus on the middle part where we have the word is, we can see that we do have an inequality phrase in there because we have the words more than. However, right after the word is, we have the number five. And whenever is is followed by a number, it translates into an equal sign. So this will become ten equals five more than a number or ten equals five plus x. And for our last example, we have ten is more than a number. So we have the words it is, but it's directly followed by an inequality phrase. We have is more than. There's no mathematical operation and there's no number right after the word is. So we know that it is going to translate into an inequality this time. So we'll use the greater than symbol to represent more than. So we have ten is greater than x. So just remember when you see that word is, pay extra careful attention to it to see if you need to translate it into an equation or and inequality.