Problem 4 - Precalculus: Mathematics for Calculus

Question

(a) The set of all points on the real line whose distance from zero is less than 3 can be described by the absolute value inequality \(|x|\) ________ . (b) The set of all points on the real line whose distance from zero is greater than 3 can be described by the absolute value inequality \(|x|\) ________ .

Step-by-Step Solution

Answer

(a) \\( |x| < 3 \\\); (b) \\( |x| > 3 \\\\).

Step 1: Understanding Absolute Value Inequality

An absolute value inequality for a set of points can be written as \( |x| < a \) or \( |x| > a \). This represents the distance from zero being either less than or greater than some value \( a \).

Step 2: Solving Part (a): Points with Distance Less Than 3

The inequality \( |x| < 3 \) describes the set of points whose distance from zero is less than 3. This inequality means that \( -3 < x < 3 \), which translates to points between -3 and 3 on the real line.

Step 3: Solving Part (b): Points with Distance Greater Than 3

The inequality \( |x| > 3 \) describes the set of points whose distance from zero is greater than 3. This inequality means that \( x < -3 \) or \( x > 3 \), which translates to points outside the interval between -3 and 3.

Key Concepts

Real Number Line

The real number line is a straight line that represents all possible real numbers. Real numbers encompass all the points along this line, including integers, fractions, and irrational numbers. This line is infinite in both directions and usually depicted with zero in the center. Every point on the line corresponds to a unique real number.

• The numbers to the right of zero are positive.
• The numbers to the left are negative.

It effectively enables visualization of numerical operations and concepts like absolute value inequalities.
Real number lines are fundamental in modeling real-world scenarios, especially when considering distances, lengths, and measurements. They are an essential concept in understanding mathematical inequalities and intervals. Understanding how numbers behave on this line aids in visualizing solutions to equations and inequalities.

Distance from Zero

"Distance from zero" is a common term used to describe the absolute value of a number. It measures how far a number is from zero on the real number line.

• The absolute value of a number is always non-negative.
• For any real number \(x\), its absolute value is represented as \(|x|\).

For example, if you consider the number 5, its distance from zero is 5 units on the number line to the right. Similarly, the number -5 also has a distance from zero of 5 units but to the left. This non-directional nature makes absolute value inequalities useful for expressing distances without regard to the direction.
When an absolute value inequality is expressed as \(|x| < a\), it implies that the points are within \(-a\) and \(a\) on the number line. If it’s \(|x| > a\), the points are outside the interval formed by \(-a\) and \(a\).

Interval Notation

Interval notation provides a way to describe sets of numbers on the real number line, focused either internally or externally relative to given boundaries.

• Open interval, denoted as \((a, b)\), includes all numbers between \(a\) and \(b\) but not the endpoints.
• Closed interval, denoted as \([a, b]\), includes all numbers between and including \(a\) and \(b\).
• Open-closed \((a, b]\) or closed-open \([a, b)\) forms include one endpoint.

In the context of absolute value inequalities:

• \(|x| < 3\) implies the points \(-3 < x < 3\), using the open interval \((-3, 3)\) to indicate the set of numbers between -3 and 3.
• \(|x| > 3\) involves using two separate intervals \((-\infty, -3)\) and \((3, \infty)\), where the solutions exist outside the interval encompassing -3 to 3.

This notation is efficient and widely used to concisely describe mathematical sets and solutions.