Interval notation provides a compact way to express solution sets, particularly useful for inequalities. This notation uses parentheses and square brackets to indicate whether endpoints are included or excluded from the set. Understanding how to convert inequalities into interval notation is essential for effectively communicating mathematical solutions.

When dealing with closed intervals, such as \(0 \leq x \leq 5\), the endpoints are included in the set. In interval notation, this is represented as \([0, 5]\). The square brackets indicate that both endpoints, 0 and 5, are part of the solution. When graphing this interval on a number line, closed circles are used at the endpoints to signify their inclusion, and a line connects them to show all values in between are included.

In contrast, open intervals occur when the inequalities do not include the endpoints, such as \(0 < x < 5\). This is expressed in interval notation as \((0, 5)\), using parentheses to denote that 0 and 5 are not included in the set. On a number line, open circles are placed at the endpoints, indicating that these values are excluded, while the line connecting them shows all values in between are included.

There are also half-open (or half-closed) intervals, which combine both types of endpoints. For example, the inequality \(0 \leq x < 5\) is represented as \([0, 5)\) in interval notation. Here, the square bracket indicates that 0 is included, while the parenthesis indicates that 5 is not. When graphed, a closed circle is placed at 0 and an open circle at 5, with a line connecting them to show the range of values included.

Lastly, when dealing with inequalities that extend indefinitely, such as \(x \geq 3\), the interval notation is \([3, \infty)\). The square bracket indicates that 3 is included, while the parenthesis around infinity, \(\infty\), signifies that it is not a hard endpoint. Infinity represents an unbounded limit, so it cannot be included in the set. When graphed, a closed circle is placed at 3, and the line extends to the right towards infinity.

Understanding these concepts allows for clear and concise communication of mathematical ideas, particularly when solving inequalities and expressing their solutions in interval notation.