Properties of Inequalities 05:35 minutes

Video Transcript

Transcript Properties of Inequalities

Jackson's a wildlife tracker in Tanzania specializing in crocodiles. He's been tracking these creatures for a while and notices that when a croc has a choice of two antelopes for supper, the croc'll always go after the bigger one. Since the crocodile always wants the larger antelope, Jackson thinks the croc may understand inequalities and their properties. This is the simplified version of what Jackson sees: Jackson thinks the smaller antelope weighs about 90 lbs and the larger antelope, about 142 lbs. Mathematically, we can represent this with an inequality symbol because 90 is less than 142. Remember that inequality symbols always open towards the bigger number, just like the crocodile will open its mouth to the larger prey. It’s currently a plentiful season on the savanna, filled with lush vegetation. The antelopes have been getting plenty of food and have both put on 20 lbs. Simplified, we can see that 110 is less than 162. This is the Addition Property of Inequality. Here, we notice how we didn't change the inequality symbol, since we added the same amount to both sides of the equation. But, during the dry season, the antelopes lose 30 lbs. each. This is the Subtraction Property of Inequality. Here, we also see the the inequality remains true when we subtract the same amount from both sides. Oh no! It looks like the dry season has continued. The two antelopes lost an additional one eighth of their weight, which means they maintained seven eighths. Therefore, we can multiply both sides by seven eighths. This can also be written as 7 times 80 divided by 8 is less than 132 times 7 divided by 8. We can simplify our calculations on both sides first by dividing out the greatest common factors. The GCF of 80 and 8 is 8 and the GCF of 132 and 8 is 4. Now that we've reduced the fractions using their GCFs, the calculations look much easier! This inequality further simplifies to 70 is less than 115.5. This is called the Multiplication and Division Property of Positive Number Inequalities. Notice that the less than symbol remained the same when we multiplied both sides by 7 and divided both sides by 8. Let's look at this inequality on a number line. We know that 70 is less than 115.5. Furthermore, we know what happens when multiplying or dividing by positive numbers, but let’s take a look at what happens when we multiply both sides of an inequality by negative 1. This results in negative 70 is less than negative 115.5. Does that make sense? Looking at the number line, we can see that negative 70 is actually a larger number than negative 115.5. So, is there something we can do to make this statement true? We can change the inequality symbol. This is the Multiplication and Division Property of Negative Number Inequalities. Multiplying a number by negative number creates opposites. So, you have to remember to always flip the inequality symbol when multiplying by a negative number. We can double-check by reading the statement as a question. Is negative 70 greater than negative 115.5? Yup! We have a true mathematical sentence, ladies and gentlemen! Let's look at another problem. Say we have two numbers, negative 4 and positive 6. Looking at the relationship we can say that negative 4 is less than 6. What happens if we divide both sides by negative 2? The sign of each number changes and the numbers become half their original size. Negative 4 divided by negative 2 gives us 2, and 6 divided by negative 2 is negative 3. Since the signs of both numbers have changed, we have to flip the inequality symbol as well. This is the Division Property of Negative Number Inequalities. To check, let's read the statement as a question. Is 2 greater than negative 3? Yes it is. So we're right. Yippee! Wow! We've learned a lot, so let's look at this summary table to help review everything. When we add or subtract both sides of an inequality using the same number, we keep the original inequality symbol. When we multiply or divide both sides by a positive number we must also keep the inequality symbol. However, when we multiply or divide both sides of an inequality by a negative number, we have to flip the inequality symbol. Okay, getting back to Jackson. Hmm...it appears as though the crocodile is no longer interested in our thin antelopes. Hey...what’s that crocodile looking at?