Problem 60 - Elementary Algebra

Question

Determine the inequality given the answers expressed in interval notation. $$ (-\infty, 4) \cup[8, \infty) $$

Step-by-Step Solution

Answer

The inequality is \(x < 4\) or \(x \geq 8\).

Step 1: Analyze Interval Notation

The interval notation \((-\infty, 4) \cup [8, \infty)\) represents two separate sets of numbers. The first set, \((-\infty, 4)\), includes all numbers less than 4, but not 4 itself. The second set, \([8, \infty)\), includes all numbers greater than or equal to 8.

Step 2: Translate Intervals to Inequalities

For the interval \((-\infty, 4)\), it corresponds to the inequality \(x < 4\). For the interval \([8, \infty)\), it corresponds to the inequality \(x \geq 8\).

Step 3: Combine Inequalities

The original interval notation indicates a union of two sets. Therefore, the complete inequality is a combination of the two: \(x < 4\) or \(x \geq 8\). This combines to form the inequalities \(x < 4\) or \(x \geq 8\).

Key Concepts

Interval Notation

Interval notation is a way of writing subsets of the real number line. It simplifies expressing the set of numbers within a particular range.
To use interval notation effectively, we should know:

• Open intervals: Denoted with parentheses, like \((-\infty, 4)\), meaning that the numbers in the set do not include the endpoint (4 in this case).
• Closed intervals: Denoted with brackets, like \([8, \infty)\), where the set of numbers includes the endpoint (8 in this example).
• Infinity (\(\infty\)): Used to describe intervals that extend beyond all finite numbers, utilized in both open and closed intervals when describing infinite sets.

Interval notation is a compact form of expression, making it convenient in mathematics.

Union of Sets

The union of sets is a fundamental concept in set theory, often represented with the symbol \(\cup\). It involves combining all elements from two or more sets to form a new set that contains all possible elements.
For instance, when analyzing \((-\infty, 4) \cup [8, \infty)\), we're identifying all numbers that are either less than 4 or at least 8.
It is key to remember:

• The union represents an 'or' condition in logical terms.
• It does not repeat elements; rather, it includes all unique elements from each specified set.

Using unions efficiently enables mathematicians to define and solve more complex problems involving multiple conditions.

Solving Inequalities

Solving inequalities involves finding the solution set for variable expressions constrained by inequality signs like <, >, ≤, or ≥.
The process includes several steps:

• Identify the type of inequality—whether it is strict (\(<\) or \(>\)) or inclusive (\(\leq\) or \(\geq\)).
• Isolate the variable on one side, similar to solving equations. This may involve addition, subtraction, multiplication, or division, but remember that multiplying or dividing by a negative number reverses the inequality sign.
• Express the solution: Once isolated, the inequality solution can often be expressed in both interval notation and verbalized as an appropriate inequality expression.

Understanding inequalities is crucial for exploring mathematical relationships and relevancies beyond straightforward equalities.

Number Sets

Number sets encompass a variety of categories that classify numbers according to specific properties. In the context of inequalities and interval notation, understanding number sets is essential to forming accurate interpretations.
Some primary number sets include:

• Natural numbers (\(\mathbb{N}\)): Counting numbers starting from 1.
• Whole numbers: Similar to natural numbers but starting from 0.
• Integers (\(\mathbb{Z}\)): These include all whole numbers, both positive and negative, including zero.
• Rational numbers (\(\mathbb{Q}\)): Numbers that can be expressed as a fraction of two integers.
• Real numbers (\(\mathbb{R}\)): Consist of all rational and irrational numbers, encompassing most numbers encountered in standard mathematics.

Recognizing the properties of these number sets allows for better understanding and accurate depiction of mathematical concepts, including solving and representing inequalities.