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Interval notation is a streamlined way to express a range of numbers and is commonly used when describing solutions to inequalities. It uses specific symbols to convey whether endpoints are included in the set.

• Brackets \([, ]\) indicate that an endpoint is included in the interval. These are often referred to as closed intervals.
• Parentheses \(, )\) suggest that the endpoint is not part of the set, known as open intervals.

For example, the set of numbers greater than or equal to 12.5 is expressed in interval notation as \[12.5, \infty)\]. The square bracket \[\] shows 12.5 is included, while the parenthesis \()\) suggests infinity is not included. Infinity \(\infty\) is always paired with a parenthesis because it is not a specific, reachable number.
Converting inequalities to interval notation helps us understand the range and scope of possible solutions in a concise form.

Graphing on a number line provides a visual representation of an interval or an inequality. This makes it easier to understand at a glance the range of values included. Here are the basic steps to graph an interval such as \[12.5, \infty)\]:

• Identify the starting point of the interval, which here is 12.5.
• Place a closed dot or filled circle on 12.5 to signify its inclusion in the set.
• Draw a line or an arrow extending to the right from this point, indicating all values greater than or equal to 12.5.

This process illustrates how number lines can simplify the understanding of the intervals and are instrumental in teaching concepts of greater than or less than, and in this case, greater than or equal to.
A clear graph helps in visually affirming the information presented in both set builder and interval notation.

Inequalities express a relationship between two values or expressions, stating that one is larger or smaller than the other. They are fundamental in mathematics for showing ranges of solutions rather than exact values.
An inequality like \(x \geq 12.5\) indicates all numbers \(x\) that are either equal to 12.5 or greater. In comparison with equations, inequalities allow for multiple solutions.
There are several symbols used in inequalities to describe these relationships:

• \(>\) Greater than
• \(<\) Less than
• \(\geq\) Greater than or equal to
• \(\leq\) Less than or equal to

Understanding inequality symbols is crucial for interpreting and working with them in mathematical problems. They highlight the need to understand whether boundaries are soft (not included) or hard (included), which directly translates into different solutions and representations in set builder and interval notations.