Numerical Inequalities

In mathematics a numerical inequality is a statement that compares two numbers or algebraic expressions and indicates whether one is greater or smaller than the other. In short, it is simply an inequality.

More broadly, an inequality compares two elements of an ordered set and identifies which of the two is larger or smaller.

Here is a basic example of an inequality:

$$ 4 < 6 $$

This statement asserts that the expression on the left-hand side (LHS) is less than the expression on the right-hand side (RHS).

It is read as "four is less than six".

Note. Numerical inequalities are fundamental tools throughout mathematics, especially in problem solving, where quantitative comparisons are essential.

The main relational symbols used to express inequalities are:

">" meaning "greater than"
"<" meaning "less than"
"≥" meaning "greater than or equal to"
"≤" meaning "less than or equal to"

The first two symbols represent a strict inequality, where equality is excluded.

The latter two represent a non-strict inequality, where equality is permitted.

Note. When two inequalities use the same relational symbol, they are said to have the "same direction" or the "same orientation". For example: $$ 4 < 9 \qquad 5 < 7 $$ If the symbols point in opposite directions, the inequalities have "opposite direction" or "opposite orientation". For example: $$ 4 < 9 \qquad 6 > 3 $$

Any inequality can be rewritten using either the greater-than or the less-than symbol.

For example, consider the statement "eight is greater than four":

$$ 8 > 4 $$

We can express exactly the same relationship by reversing the sides and using the less-than symbol:

$$ 4 < 8 $$

This alternative form is read as "four is less than eight".

Both versions encode the same inequality.

Note. As with equalities, the portion of the statement to the left of the symbol is the left-hand side (LHS), and the portion to the right is the right-hand side (RHS). $$ \underset{\text{LHS}}{4} < \underset{\text{RHS}}{8} $$

Inequalities with Variables

When an inequality contains one or more unknown variables, it is called an inequation, or more generally in English, an inequality in a variable.

For example, the following inequality includes an algebraic expression with one unknown on the left-hand side:

$$ 4x + 1 < 9 $$

For this reason, it is classified as an inequality in the variable \( x \).

Solving an inequality means determining all values of the variable for which the inequality is true.

Example. The solution set of the inequality above consists of all values of \( x \) that are less than two. We apply the usual algebraic rules for inequalities: $$ 4x + 1 < 9 $$ $$ 4x + 1 - 1 < 9 - 1 $$ $$ 4x < 8 $$ $$ \frac{4x}{4} < \frac{8}{4} $$ $$ x < 2 $$

And so on.