অসমতা সমাধান: ক্রমের ট্রাঞ্জিটিভ বৈশিষ্ট্য

Transitive Property of Order

You might remember the transitive property of equality. It tells us that if $A = B$ and $B = C$, then $A = C$. The transitive property of order uses the same idea, except with $<$ or $>$ signs instead of $=$ signs. Since we can compare numbers that are close to each other or far apart, this property has a lot more power.

Think about it in terms of words. If we know that $A$ is bigger than $B$, and $B$ is bigger than $C$, we know that $A$ is bigger than the thing that is bigger than $C$. So that means that $A$ must be bigger than $C$, too.

Here's a visual. We have three buildings, $A, B,$ and $C$. Building $A$ is taller than $B$ and so has more stories. Also, building $B$ is taller than building $C$ and has more stories. The $\color{29a603}{\text{transitive property of order}}$ tells us something very useful. Building $A$ has more stories than building $C$.

Better yet, this also works in reverse with $<$ signs instead of $>$ signs. The property is stated as follows:

If $A < B$ and $B < C$ then $A < C$.

It's actually quite easy to construct these inequalities. The trichotomy property says that for any two numbers $A$ and $B$, exactly one of the following is true.

$$A = B$$ $$A < B$$ $$A > B$$

The same is true for $B$ and $C$, or any variables we might pick. This means the ability to use the Transitive Property of Order is common.

Examples

Here are a couple examples of using the transitive property of order. We'll be building lots of inequalities!

Example: Let $x$ be a variable such that $x < 7$. We don't know the value of $x$, and we aren't going to try to find it. There are other scenarios where we solve inequalities. Instead we realize that $7 < 12$. This can be seen by counting up to $12$ or by placing both numbers on the number line. Either way, the Transitive Property of Order tells us $x < 12$ as well.

Better yet, we can pick any value of $x$ that is less than $7$, and it is automatically less than $12$. That's what our special property told us.

And finally, let's do a similar example using the $>$ sign instead of the $<$ sign.

Example: Let $y$ be a variable such that $y > -4$ (yes, we can use negative numbers). We can't find the value of $y$, but we can get other information. Using a number line, we can conclude that $-4 > -8$. Thus the Transitive Property of Order tells us $y > -8$.

Better yet, we can pick any value of $y$ that is greater than $-4$, and it is automatically greater than $-8$. That's what our special property told us.

Statement of the Property

The transitive property of order for inequalities is: $$\text{If }A<B\text{ and }B<C\text{, then }\color{29a603}{A<C}.$$

This is for any variables or numbers $A, B, C$. We don't even have to know their exact values!

We can translate this into a non-textbook statement with more words. Have a look.

If $A$ is less than $B$ and $B$ is less than $C$, then we conclude that $\color{29a603}{A \ \text{ is less than } \ C}$.

In other words, we can basically cut out the inequalities involving the middle value ($B$). What's left is a statement about only $\color{29a603}{A \ \text{ and } C}$.

Another Way to Write It

The transitive property we just talked about uses $<$ signs. There is an almost identical statement using $>$ signs. We state it below.

$$\text{If }A>B\text{ and }B>C\text{, then }\color{29a603}{A>C}.$$

In words it means the following.

If $A$ is greater than $B$ and $B$ is greater than $C$, then we conclude that $\color{29a603}{A \ \text{ is greater than } \ C}$. Once again $A, B,$ and $C$ are numbers or variables. Our outcome is a statement only about $A$ and $C$. At the end of the day we forget about $B$.

Examples

Let's take a look at a few examples of how the transitive property of order works with inequalities:

Alisha gets paid more than Ben, and Ben gets paid more than Simeon.

We can write this as two separate inequalities, and then combine them into one larger inequality. This is also an example of constructing inequalities from words.

Alicia gets paid more than Ben, so: $$\text{Alisha} > \text{Ben}$$ Ben gets paid more than Simeon, so: $$\text{Ben} > \text{Simeon}$$ Thus, we know that: $$\text{Alisha} > \text{Ben} > \text{Simeon}$$

We haven't used the transitive property of order quite yet. That's coming up next.

We can figure out that $\text{Alisha} > \text{Simeon}$, so Alisha gets paid more than Simeon.

Let's look at one more example, this time presented as a word problem.

Tomasz can't remember the ages of his aunts and uncles. What he does know is that his Aunt Tasha is younger than his Uncle Peter. He also knows that Aunt Marlene is the oldest, and that Aunt Harriet is older than Uncle Peter.

Start by writing the inequality that represents this information.

Tasha is younger than Peter, so we know: $$\text{Tasha} < \text{Peter}$$ Harriet is older than Peter, so we know: $$\text{Peter} < \text{Harriet}$$ Marlene is older than everybody, so: $$\text{Harriet} < \text{Marlene}$$ We can put that all together, using the transitive property, to get: $$\text{Tasha} < \text{Peter} < \text{Harriet} < \text{Marlene}$$

Note that you could also start with Marlene and go to Tasha using "greater than" signs instead. That expression is equivalent to the one we have.

Here we have a bunch of inequalities in a single line. This is sometimes called an inequality chain.

If we wanted to apply the transitive property of order, we could say Tasha is younger than Harriet. Because of this property, we don't even need to mention Peter.

(Video) Lesson 25: Properties of Inequalities Part 1

by AIC ASIA

This video by AIC ASIA covers properties of inequalities, including the transitive property of order.

$\large{\text{Summary}}$

The transitive property of order is very similar to the transitive property of equality.

The transitive property of equality states:

$$\text{If } a=b \text{ and } b=c, \text{ then } a=c$$

We use the same train of thought to describe inequalities with the transitive property of order.

$$\text{If } a<b \text{ and } b<c, \text{ then } a<c$$

Basically, we took the first property and replaced the $=$ with $<$. Believe it or not, this new property is still true!

Let's look at it with numbers instead of variables.

We know that $1$ is less than $4$. We also know that $4$ is less than $7$. The transitive property of order lets us know that $1$ is less than $7$ without us needing to check each case individually. This becomes very important when we solve algebraic inequalities. Let's look at a basic one.

Given $a < 10$, is $a$ greater than or less than $15$?

We know that $a<10$ from the problem, and we know that $10<15$. This means that we can say for certain that $a<15$.