Problem 44 - College Algebra

Question

Write the set in interval notation. $$ \\{x \mid x \geq 7\\} $$

Step-by-Step Solution

Answer

[7, \infty)

Step 1: Understanding Set Builder Notation

The given set is \( \{ x \mid x \geq 7 \} \). This notation describes a set of all real numbers \( x \) such that \( x \) is greater than or equal to 7. This means the set includes 7 and all numbers greater than 7.

Step 2: Converting to Interval Notation

To convert the set \( \{ x \mid x \geq 7 \} \) into interval notation, we know that the set starts at 7 and goes to positive infinity. Numbers that are included in the set are denoted by square brackets (\([\)) and numbers that are not included, such as infinity, use parentheses (\(()\)).

Step 3: Writing the Interval Notation

The interval notation for the set of numbers \( x \) such that \( x \geq 7 \) is \([7, \infty)\). Here, \([7,\) indicates that 7 is included in the set and \( \infty) \) indicates that the set extends indefinitely towards positive infinity.

Key Concepts

Set Builder Notation

Set builder notation is a useful mathematical tool that helps describe sets in a concise yet clear manner. This notation is structured using the format \( \{ x \mid \text{condition on } x \} \). Here, \( x \) represents any element of the set, and the condition specifies the rule or property that the elements must satisfy. For instance, in the set \( \{ x \mid x \geq 7 \} \), the condition \( x \geq 7 \) tells us that we're looking at all numbers \( x \) which are greater than or equal to 7.

• The vertical bar \( \mid \) effectively means "such that" or "for which."
• This notation is versatile as it can be adjusted for various mathematical contexts, like inequalities or relations.
• Set builder notation is often paired with interval notation, particularly when dealing with real numbers.

By using set builder notation, we streamline and clarify the set's description, which is especially useful when working with a wide range of numbers or complex mathematical conditions.

Greater Than or Equal To

The concept of 'greater than or equal to' is frequently encountered in mathematics, represented by the symbol \( \geq \). It combines two relationships: 'greater than' and 'equal to.' In \( x \geq 7 \), it means \( x \) can be any number that is equal to 7 or more than 7.

• This symbol is useful when describing inequalities or conditions in mathematical expressions.
• To solve an inequality like \( x \geq 7 \), we consider all numbers from 7 upwards, including 7 itself.
• 'Greater than or equal to' helps define both closed and unbounded sets in math.

Understanding this symbol is crucial when interpreting mathematical statements or when manipulating algebraic expressions, as it dictates the boundary of a set or solution.

Real Numbers

Real numbers are a broad category of numbers that include both rational and irrational numbers. They encompass almost any number you can think of, excluding imaginary numbers.

• This set includes positive numbers, negative numbers, fractions, and irrational numbers, such as \( \pi \) or \( \sqrt{2} \).
• The real number line visually represents real numbers, extending infinitely in both positive and negative directions.
• In mathematics, when we see a statement like \( x \in \mathbb{R} \), it means \( x \) is a real number.

Real numbers are foundational in everyday calculations and in theoretical math, making them one of the most important sets in the number system. Understanding real numbers helps in working with both basic and advanced mathematical concepts.

Infinity in Mathematics

Infinity is a fascinating and often complex concept in mathematics, representing an idea of boundlessness or endlessness. It's important to note that infinity itself is not a number but a concept used to describe a quantity that has no limit.

• In interval notation, infinity is denoted by either \( \infty \) (positive infinity) or \( -\infty \) (negative infinity).
• Since infinity is not a specific number, it cannot be "included" in a set, hence in interval notation, it's always paired with a parenthesis \(()\), never a bracket \([]\).
• Infinity is integral in calculus and other branches of mathematics, helping mathematicians and scientists understand behavior as numbers grow very large or very small.

Grasping the concept of infinity can greatly enhance your understanding of the mathematical universe and provide insight into problems of growth, decay, or continuity in various mathematical fields.