# Introduction and properties of inequations

An inequation is an expression of the type: $$2x+3 > 4$$

where the letter $x$ represents an amount to be determined and the whole expression could be read as: "We are looking for such a quantity that, if we multiply it by two and add three, the result is greater than four".

In the inequations, apart from the numbers and the unknowns (the $x$), we can find the following symbols:

• $= \; \rightarrow$ Equal to.
• $> \; \rightarrow$ Greater than.
• $< \; \rightarrow$ Less than.
• $\geqslant \; \rightarrow$ Greater than or equal to.
• $\leqslant \; \rightarrow$ Less than or equal to.

With these symbols we can designate the inequations and also the inequalities.

Consequently:

• An inequality is an algebraic expression in which two or more numerical values are compared.

• An inequation is an algebraic expression in which two values are compared; we can find a variable (we will call it $x$) which is expected to be solved and, in this way, we are able to find the possible values of $x$ so that it satisfies the inequation.

Consequently, we can find expressions like the following:

(1) $2=2$

(2) $3>0>-1$

(3) $-2< 5$

(4) $4\geqslant 4$

(5) $x-1\leqslant 1$

and we can transcribe them as:

(1) two is equal to two.

(2) three is greater than zero which, in its turn, is greater than minus one.

(3) minus two is less than five.

(4) four is greater than or equal to four.

(5) $x$ minus one is less than or equal to one.

In this case (1), (2), (3) and (4) are inequalities and (5) is an inequation.

Notice that the expressions (1), (2), (3) and (4) are true (the expression (5) is neither true nor false - it is necessary to determine the values of $x$ that make the expression be true).

An example of a false expression would be: $$-3> 2$$ since minus three is not greater than two, i.e., minus three is less than two.

## Basic properties

Next, we are going to see some of the properties of inequalities (and inequations). We are going to see two basic properties that satisfy the inequalities and consequently the inequations.

For this purpose we will designate $A$, $B$ and $C$ as any three numbers.

• Property 1: The numbers $A$ and $B$ always satisfy one of the following affirmations:
• $A < B$
• $A=B$
• $A > B$

The numbers $A = 4$ and $B = 8$ satisfy the first affirmation.

The numbers $A = 1$ and $B = 1$ satisfy the second affirmation.

The numbers $A = 24$ and $B =- 13$ satisfy the third affirmation.

• Property 2: This property refers to the symmetry of the inequations or inequalites:
• If $A < B \Rightarrow B > A$
• If $A > B \Rightarrow B < A$

(note: the symbol $\Rightarrow$ means "then" or "threrefore". For example, $A < B \Rightarrow B > A$, we will read it as "if $A$ is less than $B$ then $B$ is greater than $A$).

If $A = 3$ and $B = 4$ it is easy to see that $A < B$ and at the same time $B>A$.

If $A = 7$ and $B = 6$ it is easy to see that $A > B$ and at the same time $B < A$.