Problem 44 - College Algebra

Question

For the following exercises, write the set in interval notation. $$ \\{x | x \geq 7\\} $$

Step-by-Step Solution

Answer

The interval notation for \( \{ x | x \geq 7 \} \) is \([7, \infty)\).

Step 1: Understanding the Set Notation

The given set notation \( \{ x | x \geq 7 \} \) describes a set of all real numbers \( x \) such that \( x \) is greater than or equal to 7. In this notation, \( \geq \) indicates that 7 is included in the set.

Step 2: Converting to Interval Notation

In interval notation, we use brackets to indicate whether endpoints are included or not. A square bracket "\([\)" indicates inclusion, and a parenthesis "\()\)" indicates exclusion. Since the expression \( x \geq 7 \) includes 7 and extends to infinity, we write the interval as \([7, \infty)\).

Step 3: Understanding Infinity in Interval Notation

The symbol \( \infty \) means that the interval extends indefinitely. Since infinity is not a real number, we always use a parenthesis "\()\)" with infinity. Thus, the interval \([7, \infty)\) means all numbers from 7 to infinity, including 7.

Key Concepts

Set Notation

When we see a set written in set notation, what we're looking at is a description of a group of numbers that share a specific property. For example, the set \( \{ x | x \geq 7 \} \) tells us about every number \( x \) that meets the criteria \( x \geq 7 \). We use the curly braces \( \{ \} \) to enclose the set, and the vertical bar \( | \) to mean "such that." As a result, this set includes all real numbers that are greater than or equal to 7:

• \( x \) includes numbers like 7, 8, 10, and 100, but not 6.9
• The inequality \( \geq \) specifies that 7 is part of the set

When converting to interval notation, this same set will be represented using a different system of symbols.

Real Numbers

Real numbers are the backbone of most of the numbers we use every day. They include all the numbers you can find on the number line:

• Whole numbers (like -3, 0, and 7)
• Fractions (like \( \frac{1}{2} \) and -\( \frac{3}{4} \))
• Decimals (like 3.14 or -2.718)

In the context of our set notation \( \{ x | x \geq 7 \} \), when we say real numbers, we're highlighting that every point from 7 onwards, without skipping any decimal or fraction, is included. Real numbers form a continuous line, and our set captures every part of that line starting from 7.

Inequalities

Inequalities are used in mathematics to describe the relative size or order of two objects. They are symbols like \(<, \leq, \geq, eq, \) and \(>\):

• \(\geq\) means "greater than or equal to," which includes the value itself and anything larger.
• \(\leq\) stands for "less than or equal to," indicating inclusion of a value and everything smaller.

In the set \( \{ x | x \geq 7 \} \), the inequality \( \geq \) tells us that 7 is the smallest number that can be part of this set, and every number greater than 7 is also included. It's an inclusive measure, meaning the endpoint is part of the set.

Infinity

Infinity is a concept rather than a number and is used to express unboundedness or endlessness in mathematics. In interval notation, we use infinity to indicate that the set of numbers extends indefinitely:

• Infinity is denoted by the symbol \( \infty \).
• When representing intervals like \([7, \infty)\), the \( \infty \) suggests the numbers continue forever.

Crucially, because infinity isn't a number you can "reach" or include on a number line, we always use a parenthesis \( ) \) rather than a bracket \( ] \) with it. This is true whether the interval is approaching positive infinity or negative infinity.