Problem 3 - Algebra and Trigonometry

Question

Let \(S=\left\\{-2,-1,0, \frac{1}{2}, 1, \sqrt{2}, 2,4\right\\} .\) Determine which elements of \(S\) satisfy the inequality. $$ x-3>0 $$

Step-by-Step Solution

Answer

The element in set \(S\) that satisfies the inequality is \(4\).

Step 1: Understand the Inequality

The inequality given is \(x - 3 > 0\). This means we need to find values from set \(S\) that make \(x\) greater than 3 when 3 is subtracted from \(x\). This simplifies to \(x > 3\).

Step 2: Examine Elements of Set \(S\)

The set \(S\) is \(\{-2, -1, 0, \frac{1}{2}, 1, \sqrt{2}, 2, 4\}\). We need to check each element to see if it is greater than 3.

Step 3: Evaluate Each Element

- For \(-2\), \(-2 > 3\) is false.- For \(-1\), \(-1 > 3\) is false.- For \(0\), \(0 > 3\) is false.- For \(\frac{1}{2}\), \(\frac{1}{2} > 3\) is false.- For \(1\), \(1 > 3\) is false.- For \(\sqrt{2}\), approximately 1.414, \(\sqrt{2} > 3\) is false.- For \(2\), \(2 > 3\) is false.- For \(4\), \(4 > 3\) is true.

Step 4: Conclusion

Only the element \(4\) from set \(S\) satisfies the inequality \(x > 3\).

Key Concepts

Understanding Set Theory

In mathematics, **set theory** is a fundamental branch focused on studying sets, which are collections of objects. Each object in a set is called an element, and the set is often represented using curly braces. For example, in the set \( S = \{-2,-1,0, \frac{1}{2}, 1, \sqrt{2}, 2,4\} \), the numbers inside the braces are elements of the set. Sets can be finite, containing a specific number of elements, or infinite. They are crucial for organizing and analyzing groups of numbers or objects, providing the foundational language for mathematics.

Types of Sets

Several types of sets exist, such as:

• **Universal Set**: Contains all objects under consideration.
• **Empty Set**: Has no elements, written as \( \emptyset \).
• **Subset**: A set contained entirely within another set. For example, \( \{-2, -1\} \) is a subset of \( S \).

Understanding these basic ideas helps to tackle more complex problems like inequalities and conditions.

Exploring Number Sets

The concept of a **number set** refers to a set that specifically deals with numbers. Number sets are essential in mathematics to categorize numbers into groups with common properties. Our previous set \( S \) is an example of such a set.

Types of Number Sets

There are multiple key number sets that you will encounter in mathematics. Here are some of the most important:

• **Natural Numbers (\( \mathbb{N} \))**: Whole numbers starting from 1, 2, 3, and so on.
• **Integers (\( \mathbb{Z} \))**: Positive and negative whole numbers, including zero, such as -3, 0, 7.
• **Rational Numbers (\( \mathbb{Q} \))**: Numbers that can be expressed as a fraction of two integers, like \( \frac{1}{2} \).
• **Irrational Numbers**: Numbers that cannot be expressed as a simple fraction, e.g., \( \sqrt{2} \).
• **Real Numbers (\( \mathbb{R} \))**: All rational and irrational numbers.

Knowing these types helps when solving algebraic expressions and when analyzing inequalities.

Solving Inequalities

An **inequality** is a mathematical statement that shows the relationship between expressions that are not equal. For example, the inequality \( x - 3 > 0 \) requires us to find all values of \( x \) whereby \( x - 3 \) is greater than zero.

Steps to Solve Inequalities

Here’s a simplified process for solving inequalities like the one in our problem:

• **Step 1: Isolate the Variable**: Start by moving all terms involving \( x \) to one side if needed, e.g., \( x > 3 \).
• **Step 2: Examine Elements**: Evaluate each element of your set, checking if they satisfy the inequality condition. For our set \( S \), we check each number to see if it’s greater than 3.
• **Step 3: Conclusion**: Pick values that make the statement true. Here, only \( 4 \) in \( S \) satisfies \( x > 3 \).

Inequalities are versatile and apply to various problems, helping us understand which numbers fit certain criteria within a set or wider mathematical context.