Problem 43 - Precalculus

Question

Rewrite each statement using absolute value notation, as in Example 5. The distance between \(y\) and -4 is less than 1.

Step-by-Step Solution

Answer

The statement is rewritten as \(|y + 4| < 1\).

Step 1: Understand the Concept of Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. In this case, we're looking at the distance from a number, \(y\), to \(-4\). This distance is represented as an absolute value expression.

Step 2: Write the Absolute Value Expression

To express the distance between \(y\) and \(-4\) using absolute value, we use the expression \(|y - (-4)|\). This simplifies to \(|y + 4|\).

Step 3: Set Up the Inequality

The problem states that this distance is less than 1. So we set up the inequality: \(|y + 4| < 1\).

Step 4: Rewrite the Statement

Thus, the statement "The distance between \(y\) and -4 is less than 1" can be rewritten as the inequality \(|y + 4| < 1\).

Key Concepts

Inequality

An inequality is a mathematical statement that shows the relationship between expressions that are not equal. In other words, it compares two values using symbols like ">", "<", ">=", or "<=". For example, the inequality

• \(x > 3\) means that \(x\) is greater than 3.
• \(y \leq 5\) means \(y\) is less than or equal to 5.

When working with inequalities, it's important to understand that they can be solved much like equations but with extra care around multiplication or division by negative numbers, as this reverses the inequality sign. In the exercise, the statement "The distance between \(y\) and -4 is less than 1" is translated into an inequality:

• \(|y + 4| < 1\)

This shows that the difference between \(y\) and \(-4\), in absolute terms, is less than 1.

Distance on the Number Line

Distance on the number line refers to how far apart numbers are, without thinking about direction. Imagine the number line as a ruler where each point represents a number. The distance between two numbers is the absolute value of their difference. So,

• The distance between 3 and 5 on the number line is \(|3 - 5| = 2\)
• The distance between -7 and -2 is \( |-7 - (-2)| = |-7 + 2| = 5\)

In our exercise, the distance we care about is between \(y\) and -4. Using absolute value notation, this becomes \(|y - (-4)| = |y + 4|\). This expression measures precisely how far \(y\) is from \(-4\) regardless of direction.

Absolute Value Expression

An absolute value expression represents the distance a number is from zero on the number line. The absolute value of a number \(a\), denoted by \(|a|\), is always non-negative because distance cannot be negative.

• \(|5| = 5\) because 5 is 5 units away from zero.
• \(|-3| = 3\) because -3 is also 3 units away from zero.

The absolute value expression is useful when translating real-world situations into mathematical statements. For the problem "The distance between \(y\) and -4 is less than 1," we use an absolute value expression \(|y + 4| < 1\). This expression summarizes the situation into a mathematical inequality where the distance must be less than 1.