Problem 17 - Intermediate Algebra

Question

Represent each set using a graph, interval notation, and set-builder notation. See Example 1. The set of real numbers less than 14.

Step-by-Step Solution

Answer

Graph: Open circle at 14, shade left. Interval: \( (-\infty, 14) \). Set-builder: \( \{ x \mid x < 14 \} \).

Step 1: Graph the Set on a Number Line

To graph the set of real numbers less than 14, draw a number line with a point at 14. Use an open circle at 14 to indicate that 14 is not included. Shade the number line to the left of 14 to show all numbers less than 14.

Step 2: Write in Interval Notation

In interval notation, the set of real numbers less than 14 is represented by \[ (-\infty, 14) \]The parenthesis at 14 means that 14 is not included; the interval extends infinitely to the left.

Step 3: Write in Set-Builder Notation

Set-builder notation for this set is written as:\[ \{ x \mid x \in \mathbb{R}, x < 14 \} \]This reads as "the set of all x such that x is a real number and x is less than 14."

Key Concepts

Interval Notation

Interval notation is a concise way of expressing the range of numbers that a set consists of. It uses parentheses and brackets to indicate which numbers are included or excluded from the set. Parentheses, \(( )\), are used when a boundary value is not included in the set. For instance, when we have real numbers less than 14, we use \((-\infty, 14)\).

The key points to remember while using interval notation include:

• Parentheses are for values not included (e.g., \((14)\)).
• Brackets \([ ]\) are for values that are included in the set.
• "Infinity" symbols, \(\infty\) or \(-\infty\), always have parentheses as they are not real numbers that can be reached or included.

These notations are helpful for both simple and complex number ranges. It complements the visual understanding gained from a graph on the number line.

Set-Builder Notation

Set-builder notation offers a formal mathematical way to describe a set of numbers based on a common property that its elements share. It presents the numbers as a statement, using symbols like \( \{ \} \) to enclose the expression.

When expressing the set of all real numbers less than 14, we write:

• \( \{ x \mid x \in \mathbb{R}, x < 14 \} \)
  This reads as "the set of all \(x\) such that \(x\) is a real number and \(x\) is less than 14."

Symbols used to define the elements include:

• \( \mid \) or \(:\) meaning "such that."
• The set notation \( \{ \} \) encompasses the entire statement.
• \(x \in \mathbb{R}\) signifies that \(x\) is a real number.

This approach allows for a precise expression of conditions that define a set, and it's especially useful for abstract mathematics, such as when conditions are complex.

Graphing Real Numbers

Graphing real numbers on a number line is a straightforward way to visualize which numbers are included in a set. The number line is a horizontal line with markers indicating all possible real numbers. To graph a set like numbers less than 14, you:

• Identify the point where the boundary value occurs (at 14).
• Draw an open circle at 14, illustrating that 14 is not included.
• Shade the line to the left of this point to cover all numbers that are less than 14.

The open circle is crucial because it visually conveys that the boundary number (14) is the cutoff, but not part of the set. This visual tool helps in quickly understanding which numbers belong to the set. It can aid learning as it bridges number analysis with practical graphical representation.