Problem 5 - A Graphical Approach to Precalculus with Limits

Question

Using interval notation, write each set. Then graph it on a number line. $$\\{x | 1 \leq x<2\\}$$

Step-by-Step Solution

Answer

The set is \([1, 2)\) and is graphed with a solid dot at 1 and an open circle at 2.

Step 1: Understand Set Notation

We are given the set \( \{ x | 1 \leq x < 2 \} \). This notation means we are describing all numbers \( x \) such that they are greater than or equal to 1 and less than 2.

Step 2: Convert to Interval Notation

In interval notation, a square bracket \([\) denotes that the endpoint is included, while a parenthesis \(()\) denotes that the endpoint is not included. So, the set \( \{ x | 1 \leq x < 2 \} \) is represented in interval notation as \([1, 2)\). The number 1 is included, but 2 is not.

Step 3: Draw the Number Line Graph

To graph the interval \([1, 2)\) on a number line, draw a solid dot at 1, indicating that 1 is included in the set. Draw an open circle at 2, showing that it is not included. Shade the region between 1 and 2.

Key Concepts

Set Notation

Set notation is a way to express a collection of elements, often numbers, using a specific format that provides important details about them. In this concept, we interpret statements like \( \{x | 1 \leq x < 2\} \). This means that the set includes all numbers \(x\) that satisfy the condition of being greater than or equal to 1 and less than 2.
Set notation is particularly useful in mathematics for defining sets concisely. Here are some key aspects to remember:

• The vertical bar \(|\) separates the variable from the condition it must satisfy.
• The curly braces \(\{\}\) enclose the description of the set.
• Inequalities within the set description define the bounds of the set.

Understanding these basics helps you interpret the given set correctly and know how to work with it further.

Number Line Graph

A number line graph provides a visual representation of an interval or set of numbers, showing which numbers are included and which are not. To graph the set \([1, 2)\), you start by drawing a horizontal line to represent the number line.
Identify the key points from the interval and mark them accordingly:

• A **solid dot** on the number line indicates included numbers, so 1 is marked with a solid dot because it is part of the set.
• An **open circle** indicates excluded numbers, so at 2, you place an open circle to show it's not included in the set.
• **Shade** the section of the number line between these points to visually present the interval covered by the set.

This method helps you grasp the concept of the interval easily and communicate it clearly.

Inequalities

Inequalities express conditions of comparison between numbers or expressions. In this context, inequality symbols \( \leq \) and \(<\) are used to define the bounds of the set. For the set \( \{x | 1 \leq x < 2\} \), this means:

• \(1 \leq x \) tells us that \(x\) can be any number equal to or greater than 1.
• \(x < 2\) specifies that \(x\) can be any number less than 2, but not 2 itself.

These inequalities clearly define the limits of the set and help in converting to interval notation. They are crucial for understanding which values meet the criteria of a particular mathematical condition and avoiding any misinterpretation.

Graphical Representation

Graphical representation involves creating a visual interpretation of mathematical data or concepts, such as sets and intervals. When you handle graphical representations, it’s important to adhere to standards like using dots and circles on number lines, or shading areas to signify range.
For this specific case, you're representing the interval \([1, 2)\) graphically. Here's why it's efficient:

• Provides a quick, visual understanding of which values are included in the set.
• Helps to present information clearly and concisely.
• Enhances comparison and comprehension of intervals versus other numerical data.

Graphical representations in mathematics aid teachers and learners alike by breaking down complex numerical relationships or conditions into simpler and more understandable visual forms.