What Are Basic Questions About Understanding Inequalities?

[paulb203]

TL;DR Summary

Why are there greater than or EQUAL to / less than or EQUAL to signs involved when it comes to INEQUALITIES if it's supposed to be about expressions that are NOT EQUAL to one another?

Hello. I have a GCSE level (or below) question about inequalities. I got the following from BBC Bitesize https://www.bbc.co.uk/bitesize/guides/z9vkqhv/revision/1

"Inequalities are the relationships between two expressions which are not equal to one another. The symbols used for inequalities are <, >, ≤, ≥.

7>x reads as '7 is greater than x' (or 'x is less than 7', reading from right to left).

x≤−4 reads as 'x is less than or equal to -4' (or '-4 is greater than or equal to x', reading from right to left)."

I have two questions regards the above.
Are 7 and x the two expressions in the first example? Are x and -4 the two expressions in the second example?
And, why are there greater than or EQUAL to / less than or EQUAL to signs involved if this is about expressions which are NOT EQUAL to one another?
Regards the second example, for example (x is less than or equal to -4) ; what if x is -4? Then it's equal to -4 and not an 'inequality', yes/no?

 

[PeroK]

Regards the second example, for example (x is less than or equal to -4) ; what if x is -4? Then it's equal to -4 and not an 'inequality', yes/no?

No. If $x = -4$ then a) $x \le -4$ and b) $x \ge -4$.

Why do we have inequalities like $x \ge 0$? Because they are useful. For example, the equation $y = \sqrt x$ is valid for $x \ge 0$.

 

[paulb203]

Thanks, PeroK. To check I've got the very basics before I ask about your answer; does x is less than or equal to -4 mean that the value of x could be -4, or -3, or -2, etc, etc?

 

[PeroK]

Thanks, PeroK. To check I've got the very basics before I ask about your answer; does x is less than or equal to -4 mean that the value of x could be -4, or -3, or -2, etc, etc?

We use a variable such as $x$ in two related, but subtlely different, cases in mathematics:

1) As an "unknown". Here $x$ is some number that we do not know, although we may be able to work out what it is. Or, we may be talking hypothetically about some particular number. For example:

$x$ could be the length of an object in metres.

In this case, we might do some measurements that show that $x < 10$. That is then some information about our unknown number.

2) As a "variable" that is allowed to take all values in some domain. In this case, $x$ stands for each and every number in the domain. This is especially important when we talk about graphs and functions. We talk about the "x-axis", where $x$ represents each and every number along the number line. Not just one unknown number, as in case 1).

In this case, we may restrict our domain - e.g. the function $y = \sqrt x$ is only valid for $x \ge 0$. So, here $x$ represents each and every non-negative number.

It was a huge insight by Descartes, I believe, to develop the idea of the variable $x$ in case 2). The ancient mathematicians tended to see $x$ only as in case 1) and that held back the development of mathematics.

 

[berkeman]

Thanks, PeroK. To check I've got the very basics before I ask about your answer; does x is less than or equal to -4 mean that the value of x could be -4, or -3, or -2, etc, etc?

It might be best to picture the Number Line to help you visualize what the inequality means. Using the Number Line figure below, can you tell us the answer to your question?

https://d138zd1ktt9iqe.cloudfront.n...ber-line-negative-and-positive-1634275662.png

 

[pbuk]

Sometimes Bitesize oversimplifies things, particularly at the very beginning of KS3. A more complete definition would be

Inequalities are the relationships between two expressions which can have values that are not necessarily equal to one another. The symbols used for inequalities are <, >, ≤, ≥ and ≠.

 

[FactChecker]

7>x reads as '7 is greater than x' (or 'x is less than 7', reading from right to left).

x≤−4 reads as 'x is less than or equal to -4' (or '-4 is greater than or equal to x', reading from right to left)."

I have two questions regards the above.
Are 7 and x the two expressions in the first example?

Yes.

Are x and -4 the two expressions in the second example?

Yes.

And, why are there greater than or EQUAL to / less than or EQUAL to signs involved if this is about expressions which are NOT EQUAL to one another?

The reference stated the definition carelessly. They should have that the inequality symbols are $\lt$ and $\gt$. Where the equal bar is included, as in $\le$ and $\ge$, equality is also allowed.

Regards the second example, for example (x is less than or equal to -4) ; what if x is -4? Then it's equal to -4 and not an 'inequality', yes/no?

Yes, if x=-4. But $\le$ allows more than that in one symbol.

 

[pbuk]

The reference stated the definition carelessly. They should have that the inequality symbols are $\lt$ and $\gt$. Where the equal bar is included, as in $\le$ and $\ge$, equality is also allowed.

Following @FactChecker's post I'll add even more to my definition:

Inequalities are the relationships between two expressions which can have values that are not necessarily equal to one another. The symbols normally used for inequalities are $\lt, \gt, \le, \mathrm{and} \ge$; we sometimes also consider $\ne$ to be an inequality symbol. We call $\lt$ and $\gt$ (and also $\ne$ if we are using it) strict inequalities, and although it is correct to read $x \lt 4$ as "x is less than 4" we can also read it as "x is strictly less than 4". We read $x \ne 4$ as "x is not equal to 4".

You can see why Bitesize simplified it.

 

[Lnewqban]

... does x is less than or equal to -4 mean that the value of x could be -4, or -3, or -2, etc, etc?

Welcome!
This may help when you have to solve inequalities:

https://www.mathsisfun.com/algebra/inequality-solving.html

 

[Svein]

Those signs are important in negations: If we know that NOT (x<4) then we have (x≥4).

 

[DrClaude]

We call $\le$ and $\ge$ (and also $\ne$ if we are using it) strict inequalities

You made a typo here. It is < and > that are strict inequalities.

 

[PeroK]

Thanks, PeroK. To check I've got the very basics before I ask about your answer; does x is less than or equal to -4 mean that the value of x could be -4, or -3, or -2, etc, etc?

Note that $-3 > -4$. If we take $x$ to be an integer, then $x \le -4$ means $x$ could be $-4, -5, -6 \dots$

 

[pbuk]

You made a typo here. It is < and > that are strict inequalities.

Thanks, corrected.