Problem 41 - College Algebra with Corequisite Support

Question

A machinist must produce a bearing that is within 0.01 inches of the correct diameter of 5.0 inches. Using \(x\) as the diameter of the bearing, write this statement using absolute value notation.

Step-by-Step Solution

Answer

The inequality is \(|x - 5.0| \leq 0.01\).

Step 1: Understanding the Problem

The machinist needs to produce a bearing with a diameter that is within 0.01 inches of the correct diameter, 5.0 inches. This means the diameter can be 0.01 inches more or less than 5.0 inches.

Step 2: Setting Up the Absolute Value Inequality

The absolute value inequality will represent the acceptable range for the diameter. If the actual diameter is represented by \(x\), we want the difference between \(x\) and 5.0 inches to be no more than 0.01 inches. This can be written as: \[ |x - 5.0| \leq 0.01 \]

Step 3: Explaining the Absolute Value Notation

The absolute value \(|x - 5.0|\) indicates the distance between \(x\) and 5.0 on a number line. The inequality \(|x - 5.0| \leq 0.01\) means that \(x\) can vary from 5.0 by at most 0.01 inches, which suits the machinist's requirement.

Key Concepts

Understanding Inequalities

An inequality is a mathematical expression that shows the relationship between two values that are not equal. It indicates whether one value is greater than, less than, equal to, or not equal to another. In this problem, we're using an inequality to represent the acceptable range for the diameter of a bearing.

Inequalities are often expressed using symbols such as:

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

In this exercise, we use the inequality \(|x - 5.0| \leq 0.01\). This signifies that the difference between \(x\) and 5.0 must be 0.01 inches or less. Inequalities such as these are vital in real-world scenarios like manufacturing, to ensure precision.

Tolerance in Manufacturing

Tolerance refers to the permissible limit of variation in a physical dimension. It emphasizes how much a specific measurement is allowed to vary from a specified standard. In the context of machining and manufacturing, tolerance ensures parts fit together correctly and function as intended.

In our bearing example, the given tolerance was \(0.01\) inches. This means that the diameter of the bearing can either be more or less by \(0.01\) inches from the specified 5.0 inches. Tolerance is crucial because it provides flexibility in production while still adhering to precise standards.

• Ensures parts function as expected even if slight variations occur.
• Reduces waste by allowing a bit of leeway in dimensions.
• Affects cost, as tighter tolerances often demand higher precision tools.

Manufacturers need to define tolerances carefully to balance accuracy and cost.

The Role of Algebra in Problem-Solving

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities. It provides us with a powerful way to solve equations and understand relationships between variables through constants and operators.

In this exercise, algebra helps us use an absolute value inequality to express the tolerance as:\[|x - 5.0| \leq 0.01\]

• Absolute value measures how far a number is from zero, ignoring direction. Thus, \(|x - 5.0|\) represents the distance from \(x\) to 5.0.
• This helps in explaining not just exact, but acceptable ranges or limits in problems.
• Algebra allows for these relationships to be simplified, organized, and solved in a logical manner.

By employing algebra, particularly with absolute value notation, machinists can set clear and quantifiable boundaries for dimensions, ensuring precision.