When do we use $\le$ or $\ge$ sign?

Question

This might sound really dumb question but I came across this sentence from "Elementary mathematics" by Dorofeev , Potapov & Rozov that -

Yet the understanding of the sign $\le$ between concrete numbers signifies that not a specific number can be substituted in place of $x$ in the inequality $x\le 3$, which is to say that $\le$ cannot be used to relate to any numbers whatsoever.

So does this means that we can't use $\le$ or $\ge$ for anything other than variables?

Answer 1

We use $\geqslant$, and $\leqslant$ while solving equations using variables. For example 3x $\geqslant$ 6, and similarly $\leqslant$. It can also be used for concrete numbers like 4 $\geqslant$ 3, however it is understood that 4>3 so we just remove the = sign. Moreover, one can also use 4$\geqslant$ 4 but, it is pre-understood that 4=4 so we remover the > sign. In each of the case the statement is true.

Answer 2

You are quoting the book out of context (and seemingly quoting it incorrectly, as well). The passage you quote is from the opening remarks of the text, where the authors admonish the reader to make sure that they are grokking the fundamental concepts (and not just the algorithmic or computational rules). A more lengthy quote is as follows:

The student [i.e. the reader] will also recall the signs of weak inequalities $\le$ (less than or equal to) and $\ge$ (greater than or equal to). The student usually finds no difficulty when using them in formal transformations, but examinations have shown that many students do not fully comprehend their meaning.

To illustrate, a frequent answer to: "Is the inequality $2\le 3$ true?" is "No, since the number 2 is less than 3". Or, say, "Is the inequality $3\le 3$ true?" the answer is often "No, since 3 is equal to 3". Nevertheless, students who answer in this fashion are often found to write the result of a problem $x\le 3$. Yet their understanding of the $\le$ sign between concrete numbers signifies that not a single specific number can be substituted in place of $x$ in the inequality $x\le 3$, which is to say that the $\le$ sign cannot be used to relate any numbers whatsoever.

Notice that the authors are discussing a common error that students of mathematics make. They are not declaring that $2 \not\le 3$ or that $3 \not\le 3$. Instead, they are pointing out that many students, lacking a fundamental understanding of what these symbols mean, will mistakenly declare that it cannot be that $2\le 3$ because $2 < 3$.

They continue and explain this in a bit more detail:

Actually, the situation is this: by definition of the sign $\le$, the inequality $a\le b$ is considered to be true when $a<b$ and also when $a=b$. Thus, the inequality $2\le 3$ is true because 2 is less than 3, and the inequality $3\le 3$ is true because 3 is equal to 3.

From this definition of the sign $\le$ it follows that the inequality $a\le b$ is not true only when $a > b$. For this reason, the sign $\le$ may be read not only as "less than or equal to" but also as "not greater than". Thus the inequalities $2\le 3$ and $3\le 3$ are read, respectively, "2 is not greater than 3" and "3 is not greater than 3".

This second reading of the inequality might be a more useful one to keep in mind. Instead of trying to keep track of a disjunction (which requires keeping track of two statements), it might be easier to keep track of a single statement. That is, $a \le b$ is the negation of the statement $a > b$. In other words $a \le b$ means identically the same thing as $\lnot(a>b)$, i.e. "$a$ is not greater than $b$." The same reasoning applies to $a \ge b$.

Answer 3

You can use the ordering symbols for both numbers and variables. Variables, after all, are meant to represent numbers.

Read $x \le y$ as $x$ is less than OR equal to $y$. In this context, OR is the logical disjunction which evaluates to true in the case of $x<y$, true in the case of $x=y$, and false otherwise.

Then $2 \le 3$ is true, as is the simpler version $2<3$.