Problem 84 - Precalculus Student Solutions Manual 5th

Question

Write each statement as an absolute value equation or inequality. \(z\) is no less than 5 units from 4.

Step-by-Step Solution

Answer

\(|z - 4| \geq 5\)

Step 1: Understand the problem

The problem states that the variable \(z\) is no less than 5 units away from the value 4. This means the distance between \(z\) and 4 is at least 5.

Step 2: Set up the absolute value inequality

To represent the distance between \(z\) and 4 being at least 5 units, use the absolute value expression \(|z - 4| \). The inequality should reflect that the distance is no less than 5, which translates to \(|z - 4| \geq 5 \).

Step 3: Write the final inequality

Combine the information from the previous steps to write the inequality: \(|z - 4| \geq 5 \). This inequality correctly represents the distance condition given in the problem.

Key Concepts

distance interpretation

In mathematics, distance interpretation is a useful tool to understand the relationship between two numbers on a number line. The distance between two points is always non-negative, in this case, we are dealing with the distance between variable \( z \) and the number 4. When we say \( z \) is no less than 5 units away from 4, it means that the distance is at least 5 units. Imagine a number line: If our number 4 is at a certain point, we are looking for the points that are either 5 units to the right or 5 units to the left of 4. This helps in visualizing and understanding the conditions given in a problem.

absolute value expressions

Absolute value represents the distance of a number from zero on the number line, regardless of direction. Mathematically, absolute value is denoted by two vertical bars, like this: \(|z|\). It means we only care about the size of the number, not its sign. For example, \( |-5| \) and \( |5| \) both equal 5 because they are both 5 units away from 0. In the given problem, we need to find how far \( z \) is from 4, and whether that distance meets certain conditions. This can be represented by an absolute value expression as \(|z - 4|\). Here, we are checking how far any number \( z \) is from 4, treating the distance measurement as a positive value.

inequality translation

Inequality translation involves converting word problems into mathematical inequalities. For the given problem, we're told that \( z \) is no less than 5 units from 4. To translate this to a mathematical expression, we start with our absolute value expression \(|z - 4|\). The phrase 'no less than' means that the distance must be at least 5 units, which can be written as \( \geq 5 \). Therefore, the formal inequality we derive from the problem is \(|z - 4| \geq 5 \). This means that the values of \( z \) are those which are either at least 5 units greater than or 5 units lesser than 4. Such translations are crucial because they allow us to analyze and solve real-world problems using mathematical techniques.