Problem 42 - Precalculus

Question

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

Step-by-Step Solution

Answer

\(|x - 1| > \frac{1}{2}\)

Step 1: Understanding the Problem

We need to express the statement about distance between the variable \( x \) and the number 1 in terms of absolute value. The statement says that this distance exceeds \( \frac{1}{2} \).

Step 2: Define Distance Using Absolute Value

The distance between \( x \) and 1 can be written as \( |x - 1| \). Absolute value is used to measure how far a number is from zero, regardless of direction, making it a perfect tool for expressing distance.

Step 3: Translate the Inequality

The phrase "exceeds \( \frac{1}{2} \)" implies that the expression is greater than \( \frac{1}{2} \). We translate the original distance statement into the inequality: \( |x - 1| > \frac{1}{2} \).

Key Concepts

Distance in Algebra

In algebra, the concept of distance is often expressed using absolute value. Absolute value represents the distance of a number from zero on a number line, regardless of direction. This makes it an ideal tool for measuring distance between any two numbers. In our original exercise, we were tasked with determining the distance between a variable, \( x \), and the number 1.

• To find this distance, you use the expression \( |x - 1| \), which directly measures how far \( x \) is from 1.
• This method leverages the properties of absolute value because absolute values are always non-negative.

Understanding distance as an absolute value helps simplify and solve many algebraic problems by focusing solely on magnitude, not direction. This is especially useful for expressing real-world scenarios mathematically, such as navigating a route, comparing values, or even during data analysis.

Inequality Notation

Inequality notation is crucial for expressing and interpreting conditions that involve ranges rather than precise values. In our exercise, we had to convey that the distance is not merely equal to a certain number but exceeds it. This is where inequality notation shines.

• The notation \( |x - 1| > \frac{1}{2} \) indicates that the absolute difference between \( x \) and 1 is more than \(0.5\).
• This usage of 'greater than' (\( > \)) tells us that all values for \(x\) which satisfy this condition are part of the solution set.

Using inequalities allows mathematicians and students alike to describe a set of possible solutions, rather than a single pinpoint answer. As a result, it becomes an invaluable tool in solving real-world problems, where exact values are not always necessary, or available.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations. In the exercise, \( |x - 1| \) serves as an algebraic expression. It combines the variable \(x\) and the constant number 1 with subtraction, then takes the absolute value of the result.

• Expressions like this form the building blocks of algebra, allowing representation of complex relationships and functions in a simplified manner.
• Furthermore, they can be manipulated according to algebraic rules to solve equations or inequalities, expand, factor, or simplify them as needed.

A deep understanding of algebraic expressions is necessary for mastering more advanced mathematical concepts, such as functions and calculus. They enable us to generalize mathematical problems and arrive at solutions that apply universally to a multitude of scenarios.