Problem 10 - Precalculus

Question

Write the set using interval notation. $$ \\{x \mid x \neq-3,4\\} $$

Step-by-Step Solution

Answer

The set in interval notation is \((-\infty, -3) \cup (-3, 4) \cup (4, \infty)\).

Step 1: Understand the Set Description

The set \( \{x \mid x eq -3, 4\} \) consists of all real numbers except \(-3\) and \(4\). This means the set is comprehensive for all real numbers with the exception of two specific values that are excluded.

Step 2: Break the Real Number Line at Excluded Points

On the real number line, excluding \(-3\) and \(4\) creates segments. These segments are:1. From \(-\infty\) to \(-3\) (not including \(-3\)), which can be written as \((-\infty, -3)\).2. From \(-3\) to \(4\) (not including \(-3\) or \(4\)), which can be written as \((-3, 4)\).3. From \(4\) to \(\infty\) (not including \(4\)), which can be written as \((4, \infty)\).

Step 3: Combine the Intervals

Since the set is continuous, except at \(-3\) and \(4\), we combine the intervals using the union operator \(\cup\). Therefore, the set is written in interval notation as:\((-\infty, -3) \cup (-3, 4) \cup (4, \infty)\).

Key Concepts

Set Notation

Set notation is a fundamental way to describe collections of objects, which in mathematics, are often numbers. A set is commonly defined using curly braces and can be expressed as

• Listing elements {1, 2, 3}
• Defining a property {x | x > 0}

In the given exercise, the goal is to express a condition where all real numbers are included except the specific values of 3 and 4.
This can be formulated as \( \{ x \mid x eq -3, 4 \} \), meaning we are looking at all numbers \( x \) such that they aren't \(-3\) or \(4\).
Set notation allows us to succinctly state this kind of universal rule about the elements of a set.

Real Numbers

Real numbers constitute the entire set of numbers that can be found on the number line. They combine both rational numbers, like fractions and integers, and irrational numbers, which can't be written as simple fractions. Examples include:

• Whole numbers like \(0, 1, 2\)
• Fractions like \(\frac{1}{2}\)
• Decimal numbers like \(-3.33\)
• Irrational numbers like \(\sqrt{2}\)

Real numbers cover every possible value with no skips or holes between them.
In the exercise above, we're considering the entire range of real numbers but need to make sure to exclude specific numbers \(-3\) and \(4\), demonstrating how real numbers can be specific inclusion or exclusion on a broad scale of solutions.

Union of Intervals

The union of intervals is a method to combine multiple sets of numbers into a single set. When you have multiple such intervals on the real number line, the union operator \( \cup \) precisely fulfills this purpose.
Each interval clearly identifies a range between two bounds, for example:

• \((a, b)\) represents all numbers between \(a\) and \(b\) but not including \(a\) and \(b\)
• \((a, b] \) includes \(b\) but excludes \(a\)
• \((a, \infty)\) includes all numbers greater than \(a\) without an upper limit

In the mentioned scenario, we move away from the numbers \(-3\) and \(4\), forming three separate intervals:\(( -\infty, -3 )\), \((-3, 4)\), and \((4, \infty)\).
These are then combined as: \(( -\infty, -3 ) \cup (-3, 4) \cup (4, \infty)\).
This ensures we capture all numbers except those explicitly omitted.