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Inequalities - I
Inequalities. Now, we’ll have a couple videos on inequalities. So to start, I’ll say, so far, in algebra, we have focused a great deal on equations. And on what has to be true to make things equal. And that’s an important topic. But sometimes, in math, and even in real life, we need to find out whether something is bigger, smaller, or just not equal to something else.
And this is the basis of the idea of inequalities. The test uses five inequality symbols that you need to know and understand. These are the five inequality symbols. So let’s talk about these. First of all, the less than and greater than symbols. A is less than B, C is greater than D.
First thing I’ll say, is that it might be helpful to remember these, by thinking about that as say the jaws of something like a crocodile. In other words it’s open towards the bigger thing. The crocodile is hungry, so it wants to eat the bigger thing. That’s that’s wonderful little trick that they teach little kids, but it can be very helpful actually.
If you have any confusion between these two symbols. Also notice that these two, these first two have an inverse relationship. In other words, if A is less than B, that necessarily means that B is greater than A. So we can just swap the order around. And reverse the direction in which the inequality is pointing.
Now, the less than or greater than or equal to symbols. So, A is less than or equal to B. C is greater than or equal to D. And so, these are interesting, because it could be, it leaves open the possibility that the two are equal, or that one is bigger than the other. And again, these are also inverses.
If A is less than or equal to B, that necessarily means that B is greater than, or equal to A. And finally, A is unequal to B. So one, either one could be greater or less, all we know is that they don’t have the same value. So, all five of these are used in the specifications for problems.
So, at the beginning of our problem it might say something like. If x is greater than or equal to 10 or x is unequal to 5 or both and then they’ll talk about the rest of the problem. So that is just a condition they specify to make the problem work. And so any of the five symbols can appear in that specification. Now the first thing I’ll say about these is, with inequalities, it is very important not to be naive about numbers.
What do I mean by that, naive about numbers? Think about it this way. For example, the specification x is less than 5, does not mean the same thing as x is less than or equal to 4 and this is very typical, this is a very typical mistake pattern, and whats going on here, people are thinking that x can only be a number that they can count on their own fingers or in other words, people are only thinking about positive integers.
And so, this is the type of confusion that arises when people get caught in that. When they’re thinking about, all numbers are just positive integers. All numbers are numbers I can count of my fingers. In fact, when you see the word, number. Or just when you see an x. And all it’s talking about is an x, and they don’t specify anything else.
You have to think of all possible numbers. This could be positive or negative, or zero. Could be integers, fractions, and decimals. It can be any kind of number on the number line. The whole number line is open when we just have the word number or when we just see the variable x.
Now think about this. Think of all the values that would satisfy the inequality x is less than 5 and, that would not satisfy the inequality x is less than or equal to 4. Now, of course, there are no integers in that range, but there are many other numbers. So, examples include all the decimals between four and five.
All those are things that legitimately are less than 5 but they’re not less than or equal to 4. And notice that last one in particular. 4 followed by a bunch of 9’s after the decimal point. So that’s a very interesting one. Of course, that number is less than five.
And you might think well gee, it’s, it’s awfully close to five. And yes it is close to five. Here’s the mind blowing part though. How many numbers are greater than that last number, but still less than 5? And the answer, the mind blowing answer, is infinity. Between any two points on the number line, no matter how close those two points are, between those two points, there’s an absolute infinity of numbers.
And so, that means if you leave out the numbers between four and five, you’re leaving out an infinity of numbers. You’re leaving out more numbers than there are stars in the universe. So that is the magnitude of that error. So it’s very important to be discerning when you read inequalities. It’s very important not to fall in to that trap, at any point, of thinking that all possible numbers are just the numbers that you can count on your fingers.
You’ll have to use the first four inequality symbols in problem solving, the test will not have you work with the unequal sign, in problems. So when you actually have to do work for yourself, you could see greater than, or less than, or greater than or equal to, or less than or equal to. But work that you have to do with yourself, you will not just see the unequal sign.
The unequal sign might appear in the specification of problems. In that sense, you’ll need to, to know it, but you’ll not have to work with it. An algebraic statement involving one of these four symbols is known as an inequality. All of these are inequalities. With equations, we can do almost anything, we can do almost anything to one side.
As long as we did the same thing to the other side and the two sides remain equal. Equations are very easy to handle that way. What operations can we perform to both sides of inequality that preserve the statement of the inequality? So, in other words, we have an inequality say, greater than or less than pointing in a certain direction.
We want to do same, the same thing to both sides and we want to understand, is it still, are those things still greater than or less than, we have to understand that. So, first of all, we can always add the same thing to both sides or subtract the same thing both, from both sides, and the inequality remains the same. So addition and subtraction work exactly the same way with inequalities as they work with equation.
So that much is easy. So for example, if I have the inequality x plus 7 is greater than 2 and I want to get x by itself, I subtract 7 from both sides and of course I get x is less than x, sorry, x is greater than negative 5. And so that is the solution range for x. Addition and subtraction work the same way with equations as they do with inequalities.
With multiplication and division, things are a little trickier with inequalities. We can still multiply or divide both sides by any positive number. That will preserve the inequality. So if we know what we’re multiplying by is a positive, if we’re guaranteed of that, then multiplying and dividing with inequalities is exactly the same as equations, you can just do the same thing to both sides.
What’s trickier is the multiplication or division by a negative number reverses the order of the inequalities. So, for example, if I have negative x is greater than 3. Well, if I wanna get x by itself, I have to multiply by negative one. Of course, I’ll multiply both sides by negative one, but that has the effect of changing the direction in which the inequality points.
So, the x is gonna be, the negative x becomes a positive x, the three becomes a negative three, and that greater than sign has to become a less than sign. And so the final statement is x is less than negative 3. One easy way to see why this is, is to think about what happens with ordinary numbers. So take the variables out entirely and let’s just think of ordinary numbers.
So, here are two valid, legitimate, true inequalities. It is true that 7 is greater than 3. It is true that 5 is greater than negative 2. Well, let’s multiply everything by negative signs. It is also true that negative 3 is greater than negative 7. Negative three is to the right of negative seven on the number line.
And of course, negative 5 is less than positive 2. So in other words, in order to maintain true statement when we multiply by negative one, we have to reverse the direction of the inequality. So, here’s a practice problem. Pause the video, and then we’ll talk about this. So we wanted to get x by itself, so much in the same way as with equations, first we want to collect all the x’s on one side, so I’m going to subtract 2x from both sides.
Then I get 5x minus 2x is 3x. Now I’m going to subtract seven from both sides. Now I’m gonna divide by positive 3. Because I’m dividing by a positive number, that’s just like an equation. Everything else is just gonna stay the same. I can just do the division on both sides, and I get x is less than negative 3.
Now, sometimes they’ll have the solution written in this algebraic form. Sometimes they’ll write it on a number line. And of course, x is less than negative 3 is written on a number line like this. Notice there is a circle at the three, an open circle indicating that three is not a legitimate value so negative three is not one of the values in this solution. Negative four, negative five, negative six, those are legitimate values, those are things that are less than negative three.
But of course, negative 3, itself, is not less than negative 3. So negative three is not included. So the end point is not included in that region. And the purple is the allowable region. Here’s another practice problem. Pause the video, and then we’ll talk about this.
Notice, first of all, that x cannot equal 0. If we made x equal 0, then we’d be dividing by 0, and we wouldn’t have a sensible expression. Notice also that x has to be positive. Because, the fraction on the right is a positive number.
And so the only way the fraction on the left is gonna be greater than it, or equal to it, is if it’s also positive. There’s no way that it could be negative. No negative can be greater than a positive. So x absolutely has to be positive. Because all numbers are positive, we can simply cross multiply.
So, just plain old cross multiplying. We cross multiply, and we get that 6 is greater than or equal to x. Well, if we combine all these conditions together, we see that x has to be greater than 0 and less than or equal to 6. And so this would be the allowable range. Notice that there is an open circle at zero because that’s not included, and there’s a solid dot at six.
So, the left end point is not included, the right end point is included in the solution range. Here’s another practice problem. Pause the video. And then we’ll talk about this. This is tricky because this is a series of inequalities but we do this exactly the same ways as we do anything else.
First of all, we can just subtract 5 from all three of them and that’s how we get negative 4 minus 5 is negative 9, we get, we cancel the 5 in the middle and then 17 minus 5 is 12. So we have subtracted 5. Now we’re going to divide by negative 3 to get the x by itself and of course if we divide by a negative it means both those inequality signs have to flip around.
So they flip around like this. And then we get x is between. So x is, greater than or equal to negative 4, and less than 3. So, this is the solution range. And notice that we include negative four. We have a solid dot there.
We have an open circle at three because that’s not included. So the left end point is included in the solution, the right end point is not included in the solution. Just to be clear, we can have any decimal up to 3, we could have 2.9, 2.99, 2.9999999, we could have any decimal as close as we want to 3 and that would still be in the solution, we just can’t equal 3.
In summary, we can add or subtract with inequalities, exactly as we do with equations. We can multiply and divide inequalities by positive numbers. If we multiply or divide an inequality by a negative number, this reverses the direction of the inequality.
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