نابرابری 2فصل: بخش ریاضی / درس: جبر، معادلات و نابرابری ها / درس 22
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Inequalities - II
In this video we’ll talk about a few more ideas related to inequalities. First of all, suppose that more than one inequality is given in a problem. What are we allowed to do with two separate inequalities?
So, combining inequalities. If we have two equations first of all, a equals b and b equals c we can certainly combine them to get a equal c.
Technically that’s called the transitory property on ine, transitive property of ine, of equality. Much in the same way, if we know that r is less than s and s is less than t we can deduce that r is less than s is less than t and therefore r is less than t.
So that is also a transitive property. Notice that in order for the combination to work, the common term s has to be greater than one term, and less than the other term.
And that’s very, very important. If the same term is greater than both other terms, or less than both other terms, we cannot draw any conclusion.
If c is less than f and d is also less than f, we don’t know, we know that c and d are both less than f but we don’t know how they compare to each other. So for example, if f is 100.
Then c and d are both numbers less than 100 but there’s nothing telling us which one is bigger than the other. The second issue, adding inequalities. Recall that we are allowed to add or subtract any two equations.
It turns out equations are very easy in this regard. With inequalities, we need to be more careful.
Suppose we have two inequalities, a is less than b, and c is less than d. What sorts of addition and subtraction are allowed?
First of all, we can add inequalities with the same direction. In other words, with the inequalities pointing in the same direction.
So if a is greater than b, and c is greater than d. Then we can just add them together, a plus c has to be greater than b plus d, and that sort of makes sense that if we add the two big things, it’s going to be greater than the sum of the two small things.
For example, 5 is greater than 2 and 11 is greater than 8. Those are two true inequalities, we can add them when the directions are the same.
So when we add, of course, we get another true statement, 16 is greater than 10. We do not necessarily get anything sensible if we add inequalities that do not have the same direction. So, for example, these two inequalities, now we’ve flip, flipped one around, so they have opposite direction.
Adding these, well what happens, we get 13 on both sides and in fact, that’s not an inequality at all. That’s in fact an equation. And if we had picked different initial values, we could produce a sum that would either be greater than, less than or equal to. And so it means that there’s no general rule.
When you start adding inequalities where the sign is pointing in opposite directions, basically you get mathematical nonsense.
There’s no way to predict what you’re going to get. So in this case for subtracting inequalities, we cannot subtract inequalities with the same direction. So here are a bunch of inequalities.
These are valid inequalities all lined up so the inequality is in the same direction. Notice that if we subtract. Well, in different cases we could get a less than sign, an equal sign or a greater than sign.
If we subtract inequalities with the same direction we could get any of those signs so again, it’s pure mathematical nonsense.
If you subtract two inequalities with the same direction there’s no way to predict what kind of relationship will result. But in this case, we can subtract inequalities that have opposite directions.
So, if a > b and c > d, we have no way to compare the size of a minus c to the size of b minus d.
If we do big minus big and small minus small, well, it’s hard to say how that’s gonna compare.
But we can turn the latter around, make it d minus c, and then we can subtract. a minus d is greater than b minus c. So in other words, big minus small is always going to be greater than small minus big.
So this works. Think about this statement. If a is greater than b, and d is less than c, then a minus d is greater than b minus c. Lets do a numerical example.
Suppose a equals 20, b equals 15, c equals 12 and d equals 10. Clearly, 20 is greater than 15, clearly, 10 is less than 12.
And when we subtract, we get 10 is greater than 3. So notice that the resultant inequality follows the direction of the initial inequality. The one from which we subtract.
So that sets the tone that first one from which we subtract, we subtract from that something in the opposite direction, and the result, the difference, will be the same as the original.
There are no rules for multiplication and division of inequalities, again, it always results in mathematical nonsense, you cannot do it.
You might think that if a were greater than b, and c were greater than d, it would always lead to a times c is greater than b times d.
Why wouldn’t it be that big times big is always greater than in small times small? If we were guaranteed that all the numbers are positive, then that would work. But consider this example.
Negative 10 is clearly less than positive 2, and negative 8 is clearly less than positive 3. But if we multiply, we get 8, negative 8 times negative 10, and negative times negative gives us positive 80, which is greater than 6, 2 times 3 which is 6.
So this is an example where multiplying the two quote unquote smaller numbers, the negative numbers, gives a very positive, a very large product.
And so, that’s why there are no rules whatsoever for multiplication and division of inequalities. It is definitely true that any positive number is greater than any negative number.
Now that might be an obvious statement, but this can be a useful shortcut in certain circumstance. If, if we know that a number is negative, we don’t need to calculate it, its exact value. We know for a fact, that just by fact that it’s negative, it is always less than any positive number.
Also, keep in mind that adding any positive number always makes a number bigger. So for example, x plus 10 has to be bigger than x.
In fact any positive number, it can even be a fraction, x + 2 fifth has to be greater than x. Much in the same way, subtracting any positive or adding any negative makes a number smaller always.
So, x had to be greater than x minus 3. Subtracting three necessarily makes anything smaller. So, here’s a practice problem. Pause the video, and then we’ll talk about this. So, this is a very test like problem.
And in particular, it’s a test like problem, because it’s designed with some traps that would punish people who think very naively about numbers, who think that all numbers are the numbers that they can count on their fingers.
They forget about other kinds of numbers. So first of all, that first statement, that is legitimately true. That if we add two, doesn’t matter whether the number is positive or negative, a fraction, doesn’t matter. If we add two, we make it bigger.
So that is always greater than x, and because one has to be included in the answer, we can eliminate answer D, because that one doesn’t include one.
Okay, what about 2x? Is 2x always greater than x? Now, if you’re thinking about positive numbers. This is where you could run into a problem, because, of course, if you double any positive number it gets bigger. 6 is bigger than 3, 20 is bigger than 10, that sort of thing.
The problem is when we get to negative numbers. If I have negative 10, and double that, that negative 20. Negative 10 is greater than negative 20. One way to think about that. Which, which is better? Would you rather be $10 in debt, or $20 in debt.
You’re much better off financially if you’re $10 in debt, rather than $20 in debt. And so, that’s one way to think about it. Negative 10 is definitely greater than negative 20. So with negative numbers, that just doesn’t work. So two is not always true.
And so that means we can eliminate any of the answers that include two
. And so now we’re just down to A and C. Finally, x squared. Now this is particular tricky because of course with most positive numbers you square it and it gets bigger. And with negative numbers you square a negative it becomes positive.
So of course that’s bigger also. Because any positive is bigger than a negative. So it might look like, gee we’re all set. Well first of all notice that there are a couple numbers if we square them like one and zero.
We square 1 we get 1, if we square 0 we get 0 well that’s not an inequality that’s an, that’s an equality.
1 squared equals 1, it’s not greater than 1 and also, think about the decimals between 1 and 0. If we square one-half, we get one-fourth. And one fourth is less than one-half. When we square fractions, they get smaller. And so that means that this final one is not always true.
And so we can eliminate the other answers, and just by process of elimination, the only one that works is answer choice A.
In summary, we can combine inequalities, if the common term is less than one, and greater than the other.
We can combine everything in the same direction. So, a is less than b, b is less than c, we can combine that into a is less than b less than c, which tells us directly that a is less than c.
We can add inequalities in the same direction. We can subtract inequalities in opposite directions. There’s no general rule for multiplication or division of inequalities.
Any positive number is greater than any negative number. And again, that is a very handy shortcut under many circumstances. And, finally, adding a positive number makes a number greater.
Subtracting a positive makes it less.
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