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Interval notation is a way to describe a range of numbers along the number line. It is compact and efficient, making it a preferred method for expressing intervals in mathematics. In interval notation, we use parentheses and brackets to define the boundaries of a set of numbers. For example, the interval \((a, b)\) includes all numbers greater than \(a\) and less than \(b\), without including \(a\) and \(b\) themselves. Parentheses \(( )\) indicate that the endpoints are not included in the set (open interval), while brackets \([ ]\) suggest the endpoints are included (closed interval).
For instance:

• \([3, 7]\) represents numbers from 3 to 7, including 3 and 7.
• \((3, 7]\) represents numbers greater than 3 but less than or equal to 7.

When dealing with infinity (\(\infty\)), we always use parentheses because infinity is a concept, not a number, and thus cannot be included in a set of real numbers.

Real numbers are the set of all rational and irrational numbers. They can be found on the number line and include a wide range of numbers such as 0, negative numbers, fractions, and decimals. Real numbers do not include imaginary numbers or complex numbers, as they can't be represented on a single-dimensional number line.
Some key points about real numbers include:

• Rational numbers, which are numbers that can be expressed as a fraction \(\frac{a}{b}\) where \(a\) and \(b\) are integers and \(b eq 0\).
• Irrational numbers, which cannot be expressed as a simple fraction. Examples include \(\pi\) and the square root of 2.

Real numbers are crucial in defining intervals and inequalities as they represent the universe of numbers from which we generally work, particularly in elementary algebra and calculus.

Inequalities are mathematical expressions used to compare two quantities. They illustrate how one quantity is related to another, specifying whether one is greater, less than, or equal to another. The basic symbols used include:

• \(<\) which means less than.
• \(>\) which means greater than.
• \(\leq\) which means less than or equal to.
• \(\geq\) which means greater than or equal to.

Inequalities are often used along with interval notation to describe sets of solutions. For example, the inequality \(x < 6\) indicates all real numbers \(x\) that are less than 6. The corresponding interval notation is \((-fty, 6)\), and in set-builder notation, it is written as \(\{ x \mid x < 6 \}\).
Understanding inequalities helps solve problems involving ranges or limits, making it an essential concept in both algebra and calculus.