So far, we have only worked with equalities, in equations we work with scales, which is the notion of balance. In this post we will work with the opposite, that is, imbalance.

Difference between equality and inequality

To illustrate the notion of equality and inequality, look at the following sentences:

  • Letícia is 2 years older than Alexandre (equality);
  • Letícia is at least 2 years older than Alexandre (inequality).

In this case, we have different meanings. Let $L$ be Letícia and $A$ be Alexandre, we can write the sentences as:

  • $L=A+2$ (equality);
  • $L\geq A+2$ (inequality).

where:

  • $\geq$ represents the greater than or equal symbol.

Greater than

The first inequality symbol is the one greater than $>,$ which we define as:

Definition: Let a, b belong to the real numbers $a,b\in \mathbb{R},$ with $a>b,$ we say that $a$ is greater than $b$ or $a$ is strictly greater than $b$.

Example 1.

Example 1

Represent a real number $a$ greater than 10, for every $a$ a real number $(\forall a \in \mathbb{R})$.

Example 1.

Greater than or equal

The first inequality symbol is the one greater than $\geq,$ which we define as:

Definition: Let a, b belong to the real numbers $a,b\in \mathbb{R},$ with $a\geq b,$ we say that $a$ is greater than or equal to $b$.

Example 2

Represent a real number $a$ greater than or equal to 10, for all $a$ a real number $(\forall a \in \mathbb{R})$.

Example 1.

Less than

The next inequality symbol is the one less than $<,$ which we define as:

Definition: Let a, b belong to the real numbers $a,b\in \mathbb{R},$ with $a

Example 1.

Example 3

Represent a real number $a$ less than 2, for every $a$ a real number $(\forall a \in \mathbb{R})$.

Example 1.

Less than or equal

The first inequality symbol is the one greater than $\leq,$ which we define as:

Definition: Let a, b belong to the real numbers $a,b\in \mathbb{R},$ with $a\leq b,$ we say that $a$ is less than or equal to $b$.

Example 4

Represent a real number $a$ less than or equal to 2, for every $a$ a real number $(\forall a \in \mathbb{R})$.

Example 1.

These are the concepts of inequalities between numbers.