Inequality (mathematics)

Sometimes things are not the same. One number can be more than another. One number can be less.

We use these ideas to compare sizes. We can say a number is more than or equal to another. This means it could be the same size too.

If you add the same amount to both sides, the size stays the same. If you multiply by a positive number, the order stays the same. But if you use a negative number, the order flips.

Math helps us see these patterns clearly.

In math, things are not always equal. An inequality is a way to compare two things. It shows if one number is bigger or smaller than another.

We use special signs to show this. The symbol < means "less than." The symbol > means "greater than." These are called strict inequalities. This means the numbers cannot be the same.

Sometimes, numbers can be equal. We use symbols like ≤ to mean "less than or equal to." This is also called "at most." We use ≥ to mean "greater than or equal to." This means "at least."

Math rules help us work with these signs. If you add a number to both sides, the sign stays the same.

If you multiply by a positive number, the sign stays the same. But if you multiply by a negative number, the sign flips. This is a very important rule in math.

In math, things are not always equal. An inequality is a way to compare two things. It shows if one number is bigger or smaller than another.

We use special signs to show this. The symbol < means "less than." The symbol > means "greater than." These are called strict inequalities. This means the numbers cannot be the same.

Sometimes, numbers can be equal. We use symbols like ≤ to mean "less than or equal to." This is also called "at most." We use ≥ to mean "greater than or equal to." This means "at least."

Math rules help us work with these signs. If you add a number to both sides, the sign stays the same.

If you multiply by a positive number, the sign stays the same. But if you multiply by a negative number, the sign flips. This is a very important rule in math.

Inequalities help us understand how numbers sit on a line. We can use them to show if a number is much bigger than another. Engineers often use the symbol ≫ to show a very large difference. They might also use ≪ to show something is much smaller. This helps people ignore tiny numbers that do not change the answer.

Some math rules change based on the function we use. If a function always goes up, the inequality stays the same. If a function always goes down, the sign flips.

People have used different signs for a long time. In 1670, a man named John Wallis used a horizontal bar. He put the bar above the < or > signs. Later, in 1734, Pierre Bouguer used new symbols. His symbols had double horizontal bars. Mathematicians later made them simpler with just one bar. These symbols are still used in math today.

There are many special rules named after famous people. One is called the Cauchy–Schwarz inequality. Another is known as Bernoulli's inequality. Mathematicians use these to find bounds for hard problems. A bound is like a fence for a number. It tells us a number cannot go past a certain point. This is helpful when we cannot find an exact answer.

In mathematics, an inequality is a relation that compares two expressions to show they are not equal. While an equation states that two sides are exactly the same, an inequality describes a relationship of size. It is most commonly used to compare two numbers on a number line.

This comparison allows mathematicians to define ranges of possible values rather than just a single, exact point.

There are two primary categories of inequalities: strict and non-strict. A strict inequality uses the symbols < (less than) or > (greater than). In these cases, the two values cannot be equal. For example, if a < b, then a is strictly smaller than b. Non-strict inequalities include the possibility of equality. These use the symbols ≤ (less than or equal to) or ≥ (greater than or equal to). A value that is ≤ b is often described as being "at most" b, while a value that is ≥ b is described as "at least" b.

Mathematical notation for these symbols has evolved over centuries. In 1670, John Wallis used a single horizontal bar placed above the < or > signs. Later, in 1734, Pierre Bouguer introduced symbols that featured double horizontal bars, known as "less than or equal to with double horizontal bars." Over time, mathematicians simplified Bouguer's work into the single-bar symbols we recognize today, such as ≤ or ⩽. In engineering, scientists sometimes use specialized notation to show massive differences in scale. The symbol ≪ indicates that one value is "much less than" another, while ≫ means it is "much greater than." This allows researchers to neglect tiny values when making approximations.

Inequalities follow specific logical properties when you perform arithmetic. The transitive property states that if a < b and b < c, then it must follow that a < c. When adding or subtracting a constant to both sides, the inequality remains unchanged.

However, multiplication and division require more care. Multiplying or dividing by a positive constant preserves the inequality.

If you multiply or divide by a negative constant, the inequality sign must be reversed.

Functions can also be applied to inequalities, but the result depends on the function's behavior. A monotonically increasing function, such as the natural logarithm, preserves the inequality direction.

If you apply a monotonically decreasing function to both sides, the inequality relation is reversed. This is why the rules for additive and multiplicative inverses work the way they do. These rules are specific examples of applying strictly monotonically decreasing functions to an expression.

In more advanced studies, inequalities are used to define ordered fields and partial orders. A partial order is a relation that is reflexive, antisymmetric, and transitive. A total order is a specific type of partial order where any two elements in a set can be compared. Mathematicians use these structures to organize complex sets of data. In fields like linear programming, inequalities are used to define a "feasible region."

This region represents all the possible solutions that satisfy a specific set of constraints.

Many famous inequalities exist that help bound quantities when exact answers are difficult to find. These include the Cauchy-Schwarz inequality, which relates vectors in an inner product space, and the Inequality of Arithmetic and Geometric Means. Other notable examples include Bernoulli's inequality and the Triangle inequality. These tools act as mathematical fences, providing limits that a value cannot cross. By using these bounds, mathematicians can study the behavior of complex systems even when they cannot calculate a precise number.