Problem 41 - Precalculus

Question

Rewrite each statement using absolute value notation, as in Example 5. The distance between \(x\) and 1 is at least \(1 / 2\).

Step-by-Step Solution

Answer

The distance is represented by \(|x - 1| \geq \frac{1}{2}\).

Step 1: Understand the Concept of Distance and Absolute Value

The distance between two points on a number line is expressed using absolute value. In general, the distance between a number \( x \) and another number \( a \) is represented as \( |x - a| \).

Step 2: Translate Given Information into a Mathematical Statement

The problem states that the distance between \( x \) and 1 is at least \( \frac{1}{2} \). This can be translated to: \(|x - 1| \geq \frac{1}{2}\).

Step 3: Justify the Absolute Value Inequality

The absolute value \(|x - 1|\) represents the distance between \( x \) and 1. The inequality \(|x - 1| \geq \frac{1}{2}\) correctly captures the situation where the distance is at least \( \frac{1}{2} \).

Key Concepts

Distance on a Number Line

Understanding distance on a number line is an essential concept in mathematics. Imagine a straight line where numbers are evenly spaced. Each point on this line represents a number. The distance between two points on this line is how far apart those two numbers are, which can be visually measured by counting the spaces between them.
Absolute value helps express this distance as it denotes how far a number is from zero regardless of the direction. For example, the distance between 4 and 9 is 5, represented by the expression \(|9 - 4|\). It can be seen that regardless if you calculate \(|4 - 9|\) or \(|9 - 4|\), the distance remains the same because the absolute value always yields a positive outcome.
Absolute value notation is a tool to represent distance on a number line efficiently, focusing on numerical separation without considering direction.

Absolute Value Inequality

An absolute value inequality relates to mathematical statements where absolute values determine the relationship between values concerning a given boundary. It involves comparing the distance, represented by an absolute value, to a specific number using inequality symbols like \( \geq \), \( > \), \( \leq \), and \( < \).
In our exercise, the statement "distance between \( x \) and 1 is at least \( \frac{1}{2} \)" becomes \(|x - 1| \geq \frac{1}{2}\). This indicates a distance between \( x \) and 1 that is not less than \( \frac{1}{2} \)

• If we have \(|expression| \geq number\), it means the expression's distance value is equal to or greater than the number.
• Conversely, \(|expression| < number\), implies the expression's distance is less than the number.

Solving such inequalities often requires splitting them into two separate scenarios, overcoming the absolute value notation by creating two potential equations.

Mathematical Translation

Mathematical translation is the process of converting real-world statements or problems into mathematical expressions. This skill is crucial because many real-life scenarios involve some level of translation into mathematical language to solve problems effectively.
To illustrate, let's revisit the exercise: "The distance between \( x \) and 1 is at least \( \frac{1}{2} \)." Here, understanding the phrase "distance between \( x \) and 1" suggests using absolute value notation, \(|x - 1|\).
The phrase "at least" translates into \( \geq \), as it denotes a minimum threshold must be met or exceeded. Consequently, the translation of the initial statement is \(|x - 1| \geq \frac{1}{2}\).
Accurate mathematical translation allows for clear problem-solving pathways by expressing complex verbal scenarios in a precise, mathematical form.