Застосування нерівностей для порівняння дійсних чисел

(Video) Linear Inequalities

by mathman1024

Here's a video by mathman1024 where he explains what the symbols used in inequalities means, and shows some examples of them. These symbols are usually used in between different numbers.

Summary

The symbols used for inequalities are:

$$\begin{align} &<\text{ - less than}\\ &>\text{ - greater than}\\ &\le\text{ - less than or equal to}\\ &\ge\text{ - greater than or equal to} \end{align}$$

These symbols can be used to compare any two numbers. This includes whole numbers, integers, decimals, and fractions. Pick any two numbers and we can compare them using these symbols. It's just a matter of picking the right one(s).

So, what do inequalities really mean?

The video gives a quick example of an inequality: $x>5$. So a solution would be any replacement of the variable that makes the inequality true. Let's test a number and see. Plug in the number $6$. $$\begin{align} {\color{#29A603}{(6)}}&>5\\ 6&>5 \end{align}$$ So is $6$ greater than $5$? Yes it is! Therefore, $6$ is a solution to this inequality. Try some more numbers if you're still unsure. You will find that any number that's larger than $5$ will be a solution to this inequality. There's actually an infinite number of solutions to this problem! So, how do we express all the solutions if there's infinitely many of them? We can't write them all in a list; we'd be writing forever!

Well, that's what the handy number line is for. The number line will compare any two numbers. It doesn't matter how close together or far apart they are. To build upon the previous example, if we place $5$ and $6$ on the number line, $6$ will be to the right of $5$. This is a visual way of saying that $6 > 5$. If $x$ is any number obeying the inequality $x > 5$, we can use the number line in exactly the same way. The variable can be on either side of the inequality symbol.

$\large{\text{Inequalities}}$

Here are the four main inequality symbols we see in math. It's easy to get their definitions mixed up so read them carefully.

"$<$" means "less than"

For example, if I say "$x$ is less than $4$" ($x < 4$), it means that $x$ could be any number that's under $4$.

"$>$" means "greater than"

For example, if I say "$x$ is greater than $2$" ($x > 2$), that means $x$ could be any number that's above $2$.

"$\leq$" means "less than or equal to"

Inequalities refer to equations that might have whole groups of numbers as answers.

For example, if I say "$x$ is less than or equal to $2$" ($x \leq 2$), it means that $x$ could be any number that's under $2$, or $2$ itself.

Tip: The arrow-looking thing ("$\leq$" or "$\geq$") will always be pointing to the smaller number.

"$\ge$" means "greater than or equal to"

For example, if I say "$x$ is greater than or equal to $-6$" ($x \geq -6$), it means that $x$ could be any number that's above $-6$, OR it could be equal to $-6$. Put another way, if $x$ does not equal $-6$, then it must be bigger than $-6$.

Tip: Think of the arrow-looking thing ("$\leq$" or "$\geq$") as a crocodile. It's about to eat the bigger, juicier number while ignoring the smaller one. This crocodile's mouth shows exactly what we had in mind.

Practice Problems

Any of these symbols can be used in the transitive property of order. That gives us a way of building new comparisons between numbers, without always using a number line. Now, let's get some practice in.

$\large{\text{Inequality Symbols}}$

Less Than

The symbol for less than is $<$. $$a < b \text{ says that } a \text{ is less than }b$$ Any expression that comes before the less than ($<$) symbol has smaller value than the expression that comes after it.

For example, the phase ‘$x$ is less than $2$’ can be translated into a mathematical expression: $x < 2$. The similar-sounding phrase '$2$ is less than $x$' produces the inequality $2 < x$. Even though the symbols are the same, these inequalities don't mean the same thing. Order matters!

Less Than or Equal To

The symbol for less than or equal to is $\leq$. $$a \leq b \text{ says that }a\text{ is less than or equal to }b$$ Any expression that comes before the less than or equal to ($\leq$) symbol is worth less than or equal to the expression that comes after it.

For example, the phrase ‘$x$ is less than or equal to $3$’ can be translated into a mathematical expression: $x \leq 3$.

Greater Than

The symbol for greater than is $>$. $$a>b \text{ says that } a \text{ is greater than }b$$ Any expression that comes before the greater than ($>$) symbol is worth more than the expression that comes after it.

For example, the phrase ‘$x$ is greater than $-1$’ can be translated into a mathematical expression: $x>-1$. The solutions are all numbers bigger than $-1$.

Greater Than or Equal To

The symbol for greater than or equal to is $\ge$. $$a \ge b\text{ says that }a\text{ is greater than or equal to }b$$ Any expression that comes before the greater than or equal to ($\ge$) symbol is worth greater than or equal to the expression that comes after it.

For example, the phrase ‘$x$ is greater than or equal to $-1$’ can be translated into a mathematical expression: $x \ge -1$.

These inequalities are called non-strict because we allow the two sides to be equal. Similarly, the $<$ and $>$ symbols are called strict inequalities.

We know that the operator ‘$<$‘ refers to less than, so this inequality says that $x$ is less than $-3$.

The inequality states that $x$ is less than $3$. The numbers less than $3$ are $1$ and $2$.

We know that the operator ‘$\leq$’ refers to less than or equal to, so this inequality says that $x$ is less than or equal to $-2$.

The inequality states that $x$ is less than or equal to $7$. The numbers less than or equal to $7$ are $3$, $5$, and $7$.

We know that the operator ‘$>$’ refers to greater than, so this inequality says that $x$ is greater than $3$.

The inequality states that $x$ is greater than $5$. The numbers greater than $5$ are $7$ and $10$.

We know that the operator ‘$\ge$’ refers to greater than or equal to, so this inequality says that $x$ is greater than or equal to $3$.

The inequality states that $x$ is greater than or equal to $0$. The numbers greater than or equal to $0$ are $0, 1$, and $2$.