Video: One-Step Inequalities: Multiplication or Division

In this video, we will learn how to solve one-step linear inequalities by multiplication or division.

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Video Transcript

In this video, we will learn how to solve one-step linear inequalities using multiplication or division. An inequality is a mathematical sentence that contains one of the following symbols. Greater than, less than, greater than or equal to, or less than or equal to. These symbols tell us that one side of the mathematical sentence is greater than the other side. In an equation, both sides are equal to each other. And in an inequality, one side is greater than the other.

An inequality might have a variable like this. Three times 𝑥 is less than 15. And so we want to know how would we solve for this missing variable. To find this out, let’s revisit the scale that we saw on the opening screen.

If we have the inequality three 𝑥 is less than 15 on a scale, the three 𝑥s would be lighter than the 15 side. Three unknown values are less than 15. And if three unknown values are less than 15, one unknown value would be less than what?

Well, to go from three unknown values to one unknown value, we divide by three. But just like with an equation, if we divide by three on one side, we need to divide by three on the other side. 15 divided by three is five. And so we’re saying if three unknown values are less than 15, one unknown value must be less than five. On our scale, we can show that 15 is equal to three fives. And if we take off two of the 𝑥s and two of the fives, the scale will stay at the same position.

So let’s write mathematically what happened. We divide both sides by three. Three 𝑥 divided by three equals 𝑥. And 15 divided by three equals five. We found that 𝑥 is less than five. And that means we’ve solved the inequality. Solving the inequality is finding the 𝑥-values that make the inequality true, which is a really similar process to solving equations. Let’s look at another example.

Solve the following inequality. Negative two is greater than or equal to 𝑥 over 0.8. In this inequality, on the left, we have negative two. And that value must be greater than or equal to 𝑥 over 0.8. Another way to show that would be 𝑥 divided by 0.8.

Our goal is to solve this inequality. We need to find a value or values that make the inequality true. And to do that, we’ll need to isolate 𝑥. Because on the right we have 𝑥 divided by something, to get 𝑥 by itself, we need the reciprocal. We need to do the opposite. The opposite of divide by 0.8 is multiply by 0.8. And if we multiply the right-hand side by 0.8, we need to multiply the left-hand side by 0.8. 0.8 times negative two is negative 1.6.

Now 𝑥 divided by 0.8 multiplied by 0.8 would be equal to 𝑥 times one. Divide by 0.8 multiplied by 0.8 equals one. And 𝑥 times one just equals 𝑥. So we bring down our inequality symbol. And we see that negative 1.6 is greater than or equal to 𝑥.

However, this is not a very common way to write inequalities. If we want to flip this around, we can put the 𝑥 on the left. But in order to do that, we’ll need to flip the sign. One way to check is to see if the tip of the arrow is pointing in the same direction. It was pointing at the 𝑥. So it should remain pointing at the 𝑥. And then bring down the negative 1.6. It is completely true to say negative 1.6 is greater than or equal to 𝑥. But the more common notation is 𝑥 is less than or equal to negative 1.6.

In the previous two examples, we’ve already been using multiplication and division rules for inequalities. But at this point, it might be helpful to write them out so that we can see what’s happening. An inequality still holds true when both sides of the inequality are multiplied or divided by the same positive number.

The previous two examples we looked at, we were dividing or multiplying both sides of the equation by a positive number. So we say if 𝑎 is less than 𝑏 and 𝑐 is greater than zero, if 𝑐 is positive, then 𝑎 times 𝑐 is still less than 𝑏 times 𝑐. And 𝑎 divided by 𝑐 is less than 𝑏 divided by 𝑐. On a scale that would look like this, if 𝑎 is less than 𝑏, then 𝑎 times 𝑐 will be less than 𝑏 times 𝑐. And 𝑎 divided by 𝑐 will be less than 𝑏 divided by 𝑐.

Remember, this is under the condition that 𝑐 is positive. So how would we multiply by a negative? Well, multiplying something or dividing something by a negative doesn’t really translate to a scale because we would never have negative weights. So to consider an inequality where we would need to multiply or divide by a negative to solve, we’ll actually look at a number line.

Let’s consider the inequality 𝑥 is greater than or equal to two. On a number line, that would mean a circle filled in above the two and the arrow pointing to the right. 𝑥 can be two or anything greater than two. Now what would the opposite of 𝑥 be here? Negative 𝑥. If 𝑥 is greater than or equal to two and the opposite of 𝑥 is negative 𝑥. We also know that the opposite of a number is located on the number line at the same distance to zero but on the other side. 𝑥 is greater than or equal to two starts at two places to the right of zero. Its opposite would start two places to the left of zero. And its arrow would point to the left. Opposites are mirrors of each other on a number line.

And how should we write what’s represented with this orange inequality? It should show that 𝑥 is less than or equal to negative two. Let’s look at what happens to our inequalities when we do this.

To find the opposite of 𝑥, we multiplied it by negative one. And since we multiplied one side of the inequality by negative one, we did that to the other side. And we took one additional step. We had to flip the inequality symbol. From that, we get this rule. When both sides of an inequality are multiplied or divided by the same negative number, then the inequality symbol needs to change its orientation for the inequality to still hold true. And that means if 𝑎 is less than 𝑏 and 𝑐 is equal to the negative absolute value of 𝑐. That’s just a way of saying that 𝑐 is negative. Then 𝑎 times that negative value will be greater than 𝑏 times that negative value. The signs have changed. And the same would be true for division. Let’s try an example of this type.

Which of the following inequalities is equivalent to negative four 𝑥 is less than or equal to negative one? A) 𝑥 equals one-fourth. B) 𝑥 is greater than or equal to one-fourth. C) 𝑥 is greater than or equal to four. D) 𝑥 is less than or equal to one-fourth. Or E) 𝑥 is less than one-fourth.

Our inequality is negative four 𝑥 is less than or equal to negative one. And we want to rearrange this equation so only 𝑥 is on the left side. Right now, 𝑥 is being multiplied by negative four. To get 𝑥 by itself, we’ll need to do the opposite. We’ll need to divide by negative four. But if we do that on the left, we know we need to do that on the right.

And all of a sudden, we should be thinking we’re dividing by a negative and we’re working with inequalities. And that means, in order for this inequality to remain true, we need to flip the sign. Negative four 𝑥 divided by negative four equals 𝑥. And negative one divided by negative four equals one-fourth. And that means an equivalent inequality would be 𝑥 is greater than or equal to one-fourth, which is option B.

We can check and see if this is true. We’ve just said that 𝑥 can be anything that’s greater than or equal to one-fourth. We know that one is greater than or equal to one-fourth. So if we take our original inequality, negative four 𝑥 is less than or equal to negative one, and we plug in one, we should get a true statement. Is negative four less than negative one? Yes, that’s true, which means that one is a correct solution for 𝑥.

When we’re working with inequalities, I recommend checking your solutions, because what if you forgot to flip the sign? If you forgot, you would get 𝑥 is less than or equal to one-fourth. If you did that, you could’ve chosen something like negative one to check your solution. Negative one is less than one-fourth. But when you plug that in, negative four times negative one equals positive four. And positive four is not less than negative one, which would’ve told you that you’ve done something wrong. And you need to go back and check. In this case, 𝑥 must be greater than or equal to one-fourth, which is option B.

Here’s another similar example.

Rewrite 𝑥 over six is less than negative two so that only 𝑥 appears on the left-hand side.

We have 𝑥 over six is less than negative two. Before we go any further, it’s really helpful for us to think about how we multiply and divide inequalities. When we’re multiplying or dividing with positives, the signs stay the same. And we do the same operation to both sides. When we’re multiplying or dividing with negatives, the sign is flipped. And we multiply or divide by the same thing on either side.

We’re trying to get 𝑥 by itself on the left-hand side. Currently, on the left, there is 𝑥 divided by six. And to get rid of 𝑥 divided by six, we need to multiply by six. Six is a positive value, which means our sign won’t change. But we’ll still need to multiply by the same amount on both sides. 𝑥 divided by six times six equals 𝑥. And negative two times six equals negative 12. And so we see that 𝑥 is less than negative 12.

If we wanted to graph that on our number line, we would have an open circle. And the arrow would go to the left. If we want to check if this is true, we can plug in a value that is less than negative 12 into the original inequality. I’m gonna choose negative 18. I know that negative 18 is less than negative 12, and it’s divisible by six. Negative 18 divided by six is negative three. And it is true that negative three is less than negative two, which confirms 𝑥 is less than negative 12.

In this example, we’ll have to first write our inequality before we solve it.

Write an inequality to describe the following and then solve it. Negative five times a number is at least negative 45.

We’ll start by writing the inequality. We’ve got negative five being multiplied by some number, which we’ll represent with the variable 𝑥, at least negative 45. We know the other side will have a negative 45. But what symbol should “at least” represent?

We need to consider if this could be more, equal, or less. “At least” means it could be equal, but it could not be less. “At least” could also mean more. So it’s equal to or more than. And mathematically, we would represent that with a greater than or equal to symbol. If negative five times a number is at least negative 45, then negative five 𝑥 is greater than or equal to negative 45.

This is the first part of the problem, but we’ll now need to try and solve for 𝑥. Since we’re dealing with an inequality, it should immediately be on our radar that if we’re multiplying and dividing with positive values, the sign stays the same. But if we’re multiplying or dividing with a negative number, we have to flip the sign. 𝑥 is being multiplied by negative five. And to get 𝑥 by itself, to solve for 𝑥, we’ll need to divide both sides of the equation by negative five. Since we are dividing by a negative here, we must flip the inequality symbol. Negative five 𝑥 divided by negative five equals 𝑥. The sign is flipped. And negative 45 divided by negative five is nine. Negative five 𝑥 is greater than or equal to negative 45. And that means 𝑥 is less than or equal to nine.

In this example, 𝑥 is contained in the set of natural numbers. Let’s see how that effects our solution.

Given that 𝑥 is in the set of natural numbers, determine the solution set of the inequality negative 𝑥 is greater than negative 132.

We remember that this symbol that looks kind of like an N means natural numbers, which are positive integers. If 𝑥 is in the set of natural numbers, then it cannot be negative, nor can it be fractional. It must be a positive integer. If negative 𝑥 is greater than negative 132, how can we find 𝑥?

If we multiply negative 𝑥 by negative one, we would get 𝑥. But if we’re going to multiply or divide with inequalities, we need to remember that when we’re multiplying or dividing negatives, we must flip the sign. This means we would multiply both sides of the inequality by negative one. Negative 132 multiplied by negative one is 132. And then we would flip the inequality.

We now have something that says 𝑥 is less than 132. But we also know that 𝑥 is in the set of natural numbers. So first of all, that means we’ll need to use set notation, the curly brackets. And secondly, we’re only interested in the integers less than 132. 𝑥 can’t be negative, and it can’t be between whole numbers.

The smallest value 𝑥 could be would be zero. It would then be one, two, three, continuing. We can use an ellipse to represent that. And the highest value 𝑥 can be is 131. We need to be really careful here. Just because there’s 132 here doesn’t mean 𝑥 can be equal to 132. 𝑥 must be less than that. And so the largest integer that is less than 132 is 131. Under these conditions, 𝑥 can be all the positive integers between zero and 131.

We can summarise with two key points. An inequality still holds true when both sides of the inequality are multiplied or divided by the same positive number. When both sides of an inequality are multiplied or divided by the same negative number. Then the inequality symbol needs to change its orientation for the inequality to still hold true.

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