Problem 1 - Algebra and Trigonometry

Question

\(1-8=\) 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 4 satisfies the inequality.

Step 1: Understanding the Inequality

We need to find values of \( x \) in the set \( S \) such that \( x - 3 > 0 \). This can be rewritten as \( x > 3 \). We need to identify which elements in \( S \) are greater than 3.

Step 2: Evaluating Each Element

We now evaluate each element in the set \( S = \{-2, -1, 0, \frac{1}{2}, 1, \sqrt{2}, 2, 4 \} \):- \(-2, -1, 0, \frac{1}{2}, 1, \sqrt{2}, 2 \) are all less than or equal to 3.- \(4 \) is greater than 3.

Step 3: Determining the Satisfying Element

From the evaluated elements, only the number \( 4 \) satisfies the inequality \( x > 3 \).

Key Concepts

Set Theory

Set theory is a fundamental part of mathematics that deals with collections of objects, known as sets. These objects can be anything, like numbers, letters, or even other sets. The key is that everything inside a set is clearly defined and unique. For example, the given set \( S \) in this exercise contains specific numbers: \(-2, -1, 0, \frac{1}{2}, 1, \sqrt{2}, 2, \text{and} 4 \). Each element in \( S \) is distinct, and order or repetition doesn’t matter in sets. This concept of organizing data is crucial for solving problems involving inequalities.

• A set is typically denoted with curly braces: \(\{\} \).
• Elements of a set can be anything but are usually numbers when dealing with inequalities.
• In mathematical notation, a set can be represented as \( S = \{a, b, c, \ldots\} \), where \( a, b, c, \ldots \) are the elements.

Understanding the properties of sets allows us to systematically examine each element to determine which satisfy a given condition, such as an inequality.

Real Numbers

Real numbers form the backbone of mathematics and are all numbers that can be found on the number line. This includes integers, fractions, and irrational numbers. In the context of the exercise, each element of the set \( S \) is a real number, encompassing an array of different types such as integers \(-2, -1, 0, 1, 2, 4\), a fraction \(\frac{1}{2}\), and an irrational number \(\sqrt{2}\). Real numbers can potentially satisfy inequalities, like the one in the exercise.

• Real numbers include positive and negative numbers, zero, fractions, and irrational numbers.
• Intervals on the number line help us visualize which numbers satisfy a certain inequality.
• An important subset of real numbers are rational numbers, which can be expressed as fractions.

When evaluating inequalities, understanding real numbers is essential since it provides a complete picture of possibilities along the number line.

Solving Inequalities

Solving inequalities involves finding all the values of the variable that make the inequality true. For this exercise, we need to determine which elements of the set \( S \) satisfy \( x - 3 > 0 \), which simplifies to \( x > 3 \). This means we seek numbers in set \( S \) that are greater than 3. Let's break it down:

• Analyze each element in the set \( S = \{-2, -1, 0, \frac{1}{2}, 1, \sqrt{2}, 2, 4\} \).
• Compare each element to 3, focusing on whether it is greater than 3.
• Identify \( 4 \) as the only element that satisfies \( x > 3 \), since all other numbers in the set are less than or equal to 3.

Solving inequalities requires identifying which numbers from a set meet the inequality condition, allowing us to find solutions efficiently. With practice, this analytical approach becomes intuitive.