Inequalities

An inequality resembles an equation, but instead of having an equality sign \(\large =\), we use an inequality sign. This means we are not looking for a single point where both sides are equal, but rather examining when one side is greater or less than the other.

 

Symbols

There are four common inequality signs:

• \( \large < \) means "less than"
• \( \large > \) means "greater than"
• \( \large \leq \) means "less than or equal to"
• \( \large \geq \) means "greater than or equal to"

 

Example

The equation \( \large x = 5 \) has only one solution: \( \large x = 5 \).

The inequality \( \large x > 5 \), however, has many solutions: all numbers greater than 5 satisfy the inequality.

 

Different types of inequalities

Inequalities can take very different forms depending on how the unknown variable appears.

 

The simplest are linear inequalities, where the unknown appears only in the first power, for example:

 

$$ \large 2x+3 < 7 $$

 

More complex are compound inequalities, where two inequalities are combined into one condition, as in:

 

$$ \large 1 < x < 5 $$

 

Finally, there are quadratic inequalities, where the unknown appears in the second power, for example:

 

$$ \large x^2 - 4 \geq 0 $$

 

In this case, the solution is not a single number but an entire interval of values.