Problem 13 - Precalculus

Question

Write the set using interval notation. $$ \\{x \mid x \neq 0,\pm 4\\} $$

Step-by-Step Solution

Answer

The interval notation is \((-\infty, -4) \cup (-4, 0) \cup (0, 4) \cup (4, \infty)\).

Step 1: Identify the Restrictions

The given set is defined as all real numbers except 0, 4, and -4. This means that we need to split the real number line into intervals, excluding these three points.

Step 2: Write Intervals Excluding the Restricted Values

To exclude 0, 4, and -4 from the set of all real numbers, we need to construct separate intervals: - From negative infinity to -4, which is \((-\infty, -4)\).- From -4 to 0, which is \((-4, 0)\).- From 0 to 4, which is \((0, 4)\).- From 4 to positive infinity, which is \((4, \infty)\).

Step 3: Combine the Intervals Using Union

The final step is to combine these separate intervals using the union symbol, \(\cup\). The interval notation becomes: \((-\infty, -4) \cup (-4, 0) \cup (0, 4) \cup (4, \infty)\).

Key Concepts

Real Numbers

Real numbers are the foundation of mathematics and include almost every number you can think of: integers, fractions, and decimals, both rational and irrational.

• Integers are whole numbers like -3, 0, and 4.
• Fractions could be numbers like \( \frac{1}{2} \).
• Decimals might include numbers such as 0.75.
• Irrational numbers are those that cannot be written as a simple fraction, like \( \sqrt{2} \).

Understanding real numbers helps in solving equations and graphing functions. In this exercise, real numbers exclude 0, 4, and -4. This is due to the constraints or conditions mentioned.

Set Notation

Set notation is a mathematical way to describe a collection of objects or numbers, typically using braces, like this: \( \{ ... \} \). It's like grouping things together under specific rules or conditions. In our example, the set \( \{x \mid x eq 0,\pm 4\} \) tells us about all numbers \( x \) except for 0, 4, and -4.

• The vertical bar "\(|\)" stands for "such that," creating a condition for a set.
• "\( eq \)" is the symbol for "not equal to." It excludes those specific numbers.

Through set notation, the task is to describe a series of intervals on a number line.

Inequalities

Inequalities show relationships between numbers and can indicate that one number is greater than, less than, or not equal to another. When creating intervals, inequalities provide structure.

• Symbols such as "\(<\), "\(>\), and "\( eq \)" help form these phrases.
• For instance, \( x > -4 \) or \( x < 0 \) both describe infinite sets of numbers.

In our context, inequalities assist in forming open intervals, which are stretches of the number line that exclude certain points, helping avoid numbers like 0, 4, and -4.

Number Line Segmentation

Number line segmentation involves dividing a number line into specific parts or intervals. This exercise requires excluding certain numbers entirely from the number line.

• An open interval, which is marked using parentheses \((...\)), does not include the endpoints.
• In the given solution, four separate segments exclude the numbers 0, 4, and -4.

Using number line segmentation, we understand the reason behind the division at specific points, such as at -4, 0, and 4, leading us to an understanding of how unions of these intervals represent the set.