Problem 46 - Precalculus

Question

Rewrite each statement using absolute value notation, as in Example 5. The number \(y\) is less than one unit from the number \(t\).

Step-by-Step Solution

Answer

The statement is \(|y - t| < 1\).

Step 1: Understand the Problem Statement

The problem states that the difference between the numbers \(y\) and \(t\) is less than 1 unit. Our task is to express this relationship using absolute value notation.

Step 2: Recall the Definition of Absolute Value

Absolute value measures the distance of a number from zero on the number line. If \(y\) is less than one unit from \(t\), we express the mathematical distance between \(y\) and \(t\) using absolute value notation as \(|y - t|\).

Step 3: Set Up the Absolute Value Inequality

Since \(y\) is less than one unit away from \(t\), we write this condition using an inequality with absolute value: \(|y - t| < 1\). This states that the distance between \(y\) and \(t\) is less than 1.

Key Concepts

Absolute Value Inequality

An absolute value inequality is a mathematical expression that includes an absolute value and an inequality sign, like less than, greater than, less than or equal to, or greater than or equal to. When we say an absolute value inequality, we're looking at how far apart two values are on a number line and setting a condition for that distance. For instance, if you have the expression \(|y - t| < 1\), it tells us that the distance between \(y\) and \(t\) must be less than 1 unit. This is because absolute value always gives us the non-negative distance between two points. So, the inequality \(|y - t| < 1\) ensures that \(y\) is within the range of one unit from \(t\), no matter which direction you measure from. It highlights flexibility in terms of where \(y\) can be in relation to \(t\). Nevertheless, it maintains a strict boundary defined numerically by the inequality always being less than 1.Using inequalities with absolute value can effectively set constraints when dealing with variables that can fluctuate within a range, ensuring they're neither too high nor too low.

Distance on Number Line

Understanding the concept of distance on a number line is crucial, especially when dealing with absolute value. On a number line, each point corresponds to a real number, and the distance between two points is the absolute difference between these numbers.For example, consider two numbers \(y\) and \(t\). The distance between them is represented by the expression \(|y - t|\). The absolute value here ensures that the distance is always a positive number or zero.

• If \(y\) is exactly 1 unit away from \(t\), then \(|y - t| = 1\).
• If \(y\) is less than 1 unit from \(t\), then \(|y - t| < 1\).
• If \(y\) is more than 1 unit from \(t\), then \(|y - t| > 1\).

The beauty of using absolute value for distance is that it doesn't matter if you move left or right along the number line; the value of \(|y - t|\) will be the same. This symmetric property is what makes absolute value incredibly useful for evaluating distances in a straightforward and intuitive way.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, operators, and sometimes functions, that represent a particular quantity or relationship. In algebra, an expression doesn’t include an equality or inequality sign, while a mathematical sentence does. Expressions can be as simple as a single number or variable, like \(7\) or \(y\), or as complex as \(|y - t| < 1\).Let's break down the expression \(|y - t| < 1\):

• Variable Terms: \(y\) and \(t\) are variables, which means they represent numbers whose values can change.
• Operator: The minus sign (−) shows that we find the difference between \(y\) and \(t\).
• Absolute Value Notation: The vertical bars \(|...|\) depict the absolute value, ensuring our result is non-negative. They measure how far apart the numbers are, without regard to direction.
• Inequality Sign: The less than sign (<) indicates that the result of the absolute value operation should be less than 1.

Understanding mathematical expressions helps us to articulate and solve real-world problems through abstract mathematical language. Recognizing the components and relationships in expressions allows for better problem-solving and logical thinking.