Problem 46 - Calculus Single Variable

Question

Describe each set using interval notation and the notation \(\\{x: P(x)\\} .\) Use the symbol \(\cup\) where appropriate. The set of numbers with a square that lies strictly between 2 and 10

Step-by-Step Solution

Answer

The set is \((-\sqrt{10}, -\sqrt{2}) \cup (\sqrt{2}, \sqrt{10})\).

Step 1: Understanding the Problem

We are asked to find the set of numbers whose square lies strictly between 2 and 10. This means if \( x^2 \) is the square of a number \( x \), then we have the inequality \( 2 < x^2 < 10 \).

Step 2: Solve the Inequalities

We solve \( x^2 > 2 \) which gives \( x > \sqrt{2} \) or \( x < -\sqrt{2} \). Also, solving \( x^2 < 10 \) gives \( x < \sqrt{10} \) and \( x > -\sqrt{10} \).

Step 3: Find Intersection for Positive x

For positive \( x \), combine \( x > \sqrt{2} \) and \( x < \sqrt{10} \). Thus, \( x \) lies in the interval \( (\sqrt{2}, \sqrt{10}) \).

Step 4: Find Intersection for Negative x

For negative \( x \), combine \( x < -\sqrt{2} \) and \( x > -\sqrt{10} \). Thus, \( x \) lies in the interval \( (-\sqrt{10}, -\sqrt{2}) \).

Step 5: Combine Intervals

The solution involves combining the intervals for positive and negative \( x \). Therefore, the solution set in interval notation is \( (-\sqrt{10}, -\sqrt{2}) \cup (\sqrt{2}, \sqrt{10}) \).

Step 6: Write in Set Notation

In set notation, this solution is expressed as \( \{ x: 2 < x^2 < 10 \} = (-\sqrt{10}, -\sqrt{2}) \cup (\sqrt{2}, \sqrt{10}) \).

Key Concepts

Set Notation

When using set notation, we express sets with parameters or conditions. It often uses the format \( \{ x : P(x) \} \), read as "the set of all \( x \) such that condition \( P(x) \) is true." This method enables us to precisely describe which elements belong to a set based on specific properties.

In the context of numbers, conditions related to their algebraic properties, such as inequalities, are often used. For instance, in this exercise, the condition \( 2 < x^2 < 10 \) means we're interested in finding the set of numbers \( x \) such that when squared, they fall between 2 and 10. Using set notation makes it clear which numbers satisfy this specific mathematical condition.

Inequalities

Inequalities are mathematical expressions involving symbols such as \( <, >, \leq, \geq \) to compare values. They express the relationship of being less than, greater than, less than or equal to, or greater than or equal to another value.

In solving inequalities like \( 2 < x^2 < 10 \), we have to find the range of \( x \) values that satisfy the condition. To solve these inequalities, we split them into separate parts:

• First solve \( x^2 > 2 \).
• Next, solve \( x^2 < 10 \).

By solving these, we uncover the ranges where \( x \) meets these conditions separately before finding the intersection of these solutions. Inequalities guide us to determine which elements are included in a particular set by indicating boundaries or thresholds for inclusion.

Intersection of Intervals

The intersection of intervals refers to finding the common elements shared by two or more intervals. It plays a crucial role in determining the possible values of \( x \) that satisfy multiple conditions simultaneously.

To find the intersection, you identify the overlapping regions where both sets of conditions are true. For example, with this problem, we derived two sets of intervals:

• For positive \( x \), \( (\sqrt{2}, \sqrt{10}) \).
• For negative \( x \), \( (-\sqrt{10}, -\sqrt{2}) \).

These intervals are then combined using set union notation \( \cup \) to express the full solution in interval notation, as they are separate distinct regions of \( x \) that satisfy the original inequality \( 2 < x^2 < 10 \). Intersection and union of intervals allow us to understand and present the solution within defined mathematical boundaries.

Mathematical Sets

Mathematical sets are collections of distinct objects, considered as objects in their own right. Sets are fundamental in mathematics as they provide a basic framework for defining and working with collections of numbers, shapes, or other objects.

In mathematics, a set is often written in braces, such as \( \{ x, y, z \} \), or in a descriptive form known as set-builder notation, \( \{ x : P(x) \} \). In this exercise, the set of all \( x \) such that \( x^2 \) is between 2 and 10 is our primary focus.

Working with sets involves understanding the rules for combining and intersecting them, as discussed in this problem with interval notation. Sets and the operations that we can perform on them, such as union and intersection, allow for precise articulation of mathematical concepts and problem-solving. This mirrors how real-world collections can be grouped, compared, and analyzed.