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Interval notation is a system used in mathematics to denote the range of values that lie between two endpoints on the number line. It is commonly used to specify the solution sets of inequalities and the domain and range of functions. For example, consider the interval notation \( [-4,5] \).

This interval includes all real numbers from \( -4 \) to \( 5 \) and indicates that both endpoints are included in the set, which is known as a closed interval. The square brackets \( [ \) and \( ] \) signify that the endpoints are part of the set. If an endpoint is not included, parentheses \( ( \) or \( ) \) are used, denoting an open interval. Understanding the basic rules of interval notation can be crucial in interpreting and solving problems involving continuous ranges of numbers.

Real numbers encompass all the numbers on the number line, including all the rational and irrational numbers. Rational numbers are those which can be expressed as the ratio of two integers, and irrational numbers cannot be so expressed, having non-terminating, non-repeating decimal expansions. For instance, \( 2, -1, \frac{1}{2}, \sqrt{2} \) are all real numbers. In our example \( [-4,5] \), the interval contains all real numbers from \( -4 \) to \( 5 \) including fractions and irrational numbers between them, not just integers. Any mathematical statements or expressions involving real numbers require a deep understanding of their properties and how they relate to the number line and set theory.

Inequalities are mathematical expressions involving the comparison of two quantities. They use symbols like \( < \), \( > \), \( \leq \), and \( \geq \) to signify 'less than', 'greater than', 'less than or equal to', and 'greater than or equal to', respectively. For example, the inequality \( -4 \leq x \leq 5 \) defines a set of numbers where \( x \) can be any number from \( -4 \) up to and including \( 5 \).

When converting interval notation to set-builder notation, as we do in the given exercise, we are essentially using inequalities to describe the same numeric set. The inequality expression \( -4 \leq x \leq 5 \) in set-builder notation communicates that the variable \( x \) is bounded by \( -4 \) and \( 5 \) and includes all numbers that fall within that range on the real number line.