Впорядкування дійсних чисел

Graphing Inequalities

Less Than

$$a < b \text{ says that } a \text{ is less than }b$$

We can represent this visually on a number line:

The open circle on the number line means that $2$ is not included. This means that $x$ can be any number less than $2$, but cannot be $2$ itself.

Less Than or Equal To

$$a \leq b \text{ says that }a\text{ is less than or equal to }b$$

We can represent this visually on a number line. We write all the integers on it. However, every real number has a home on the number line. We just don't label all of them.

The closed circle on the number line means that $3$ is included. This means that $x$ can be any number less than $3$, or $3$ itself.

Greater Than

$$a>b \text{ says that } a \text{ is greater than }b$$

We can represent this visually on a number line:

Inequality Symbols: Greater Than or Equal To

$$a \ge b\text{ says that }a\text{ is greater than or equal to }b$$

We can represent this visually on a number line:

The closed circle on the number line means that $-1$ is included. This means that $x$ can be any number greater than $-1$, or $-1$ itself.

Order in the Court!

Somebody, probably: "There's too many numbers! I can't keep track of them all!"

Guy who's about to invent the number line: "Hold my Capri Sun and watch this."

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While that conversation probably didn't happen, numbers do have an order to them. The larger numbers come after the smaller numbers.

Yeah yeah, I know, no duh. But, this lets us write the numbers in a nice straight line...a number line, if you will.

The numbers to the right of zero are the positives and increase as you go further right. The numbers to the left of zero are the negatives and decrease as you go further left.

To graph a number on the number line, you put a dot at the number. For example, here's the number $5$ graphed on a number line.

Now what about the reverse? Say you wanted every number but $5$. To indicate a number is not selected, we use an open circle and shade in the line everywhere else.

Last but not least, we go into the inequalities. There are four types of inequalities: less than, greater than, less than or equal to, and greater than or equal to.

Suppose we wanted every number greater than or equal to $5$. Because we have or equal to we include the number itself. This means closed circle! Also, since it's greater, we shade the line after $5$.

The numbers less than or equal to $5$ are similar, just the line goes the other direction.

Whenever you have just greater than or less than, you leave an open circle on the number. So for example, all the numbers greater than $5$ would have an open circle on the $5$.

And once again, less than $5$ is the same but with the line going the other way.

(Video) Inequalities

by mathman1024

This video by mathman1024 explains the different ways to express inequalities.

$\large{\text{Summary}}$

The first way is by using a number line to graph it.

So our problem again was $x>5$. We want to show the set of numbers bigger than $5$. So, we first put the point $5$ down on the number line. Since we are using $>$ we put an open circle. When using $<$ or $>$, use an open circle. When using $\le$ or $\ge$, use a closed circle.

Finally, shade the area that contains the solution. In our problem, the solution is any number greater than $5$, so we shade to the right (towards $+\infty$).

The second way to express the solution is with interval notation. This is what will be used most often to express the solution. Here's what our solution looks like in interval notation: $$(5,\infty)$$ Here we use parentheses $(\ \ \ )$ to denote $<$ or $>$. Parentheses are like the open circle on the number line. We use brackets $[\ \ \ ]$ to denote $\le$ or $\ge$. When dealing with infinity, $\infty$, you always use a parenthesis.

The final way to express the solution is called set builder notation. In set builder notation our solution would look like: $$\{x|x>5\}$$ The way we would say this is:

"All the values of $x$, such that $x$ is greater than $5$."

We have other resources on set builder notation if you're interested.