Problem 86 - Precalculus Student Solutions Manual 5th

Question

Write each statement as an absolute value equation or inequality. \(k\) is within 0.0002 unit of \(10 .\)

Step-by-Step Solution

Answer

\(|k - 10| \leq 0.0002\).

Step 1: Understand the Problem

The problem states that the variable \(k\) is within 0.0002 units of 10. This can be expressed as \(|k - 10|\) being less than or equal to 0.0002.

Step 2: Set Up the Absolute Value Inequality

From the problem, if \(k\) is within 0.0002 units of 10, this can be translated to:\(|k - 10| \leq 0.0002\).

Key Concepts

absolute value

Absolute value is a mathematical concept that represents the distance of a number from zero on the number line, regardless of direction. It is always a non-negative number. The absolute value of a number is denoted by two vertical bars surrounding the number, such as \(|k|\). This means that \(|k| = k\) if \k\ is positive, and \(|k| = -k\) if \k\ is negative. For example, \(|-3| = 3\) and \(|3| = 3\).

In the context of the exercise, the absolute value function \(|k - 10|\) represents the distance between the variable \(k\) and the number 10. If \k\ is within 0.0002 units of 10, then the distance between \k\ and 10 must be less than or equal to 0.0002. Therefore, we express this relationship using the absolute value inequality \(|k - 10| \leq 0.0002\).

inequality

Inequalities are mathematical expressions used to compare the sizes or values of different numbers or variables. They show that one value is less than, greater than, less than or equal to, or greater than or equal to another. Symbols used in inequalities include

• \(<\) for less than
• \(>\) for greater than
• \(\backslash leq\) for less than or equal to
• \(\backslash geq\) for greater than or equal to

In the given exercise, the inequality \(|k - 10| \leq 0.0002\) means that the distance (absolute value) between \k\ and 10 is less than or equal to 0.0002. This implies that the variable \k\ can range from \10 - 0.0002\ to \10 + 0.0002\. So, if you solve this inequality, you get:

• \k - 10 \leq -0.0002\
• \k - 10 \geq 0.0002\

In other words, the variable \k\ is restricted to the interval \([9.9998, 10.0002]\) on the number line.

variable

A variable in mathematics is a symbol used to represent unknown or changeable values. Variables are often denoted by letters such as \k, x, y,\ and so on. Variables allow us to write general formulas and equations that can solve a wide range of problems.

In the exercise, the variable \k\ represents a number whose value is not fixed and can vary. We know from the problem that \k\ is within 0.0002 units of 10, which translates to the absolute value inequality \(|k - 10| \leq 0.0002\). By understanding how to work with the variable and interpreting its constraints, students can grasp the range of possible values \k\ can have.

Analyzing variables within inequalities helps us understand and predict the behavior of the variable in real-world scenarios. For students, identifying and manipulating variables is crucial in solving equations and understanding more complex mathematical concepts.