A Quality Assurance Engineer Randomly Checks Manufactured Parts To Ensure That They Are Within 0.002 Mm Of The Desired Specifications. If The Desired Length Of The Part Is 3 Mm, Then The Compound Inequality − 0.002 ≤ X − 3 ≤ 0.002 -0.002 \leq X-3 \leq 0.002 − 0.002 ≤ X − 3 ≤ 0.002

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Introduction

In the world of manufacturing, quality assurance engineers play a crucial role in ensuring that products meet the desired specifications. One of the key tasks of a quality assurance engineer is to randomly check manufactured parts to verify that they conform to the required standards. In this article, we will explore a scenario where a quality assurance engineer checks manufactured parts to ensure that they are within 0.002 mm of the desired specifications. We will also delve into the mathematical concept of compound inequalities and how they relate to this scenario.

The Desired Specifications

The desired length of the manufactured part is 3 mm. This means that the part should be between 2.998 mm and 3.002 mm to meet the desired specifications. However, the quality assurance engineer is not just checking the length of the part, but also ensuring that it is within a certain tolerance range.

The Compound Inequality

The compound inequality −0.002≤x−3≤0.002-0.002 \leq x-3 \leq 0.002 represents the tolerance range for the manufactured part. This inequality states that the difference between the actual length of the part and the desired length (3 mm) should be between -0.002 mm and 0.002 mm. In other words, the actual length of the part should be between 2.998 mm and 3.002 mm.

Solving the Compound Inequality

To solve the compound inequality, we need to isolate the variable x. We can do this by adding 3 to all three parts of the inequality:

−0.002+3≤x−3+3≤0.002+3-0.002 + 3 \leq x - 3 + 3 \leq 0.002 + 3

This simplifies to:

2.998≤x≤3.0022.998 \leq x \leq 3.002

Interpretation of the Solution

The solution to the compound inequality 2.998≤x≤3.0022.998 \leq x \leq 3.002 represents the range of values that the actual length of the manufactured part can take. This range is between 2.998 mm and 3.002 mm, which is the desired tolerance range.

Real-World Applications

The concept of compound inequalities has numerous real-world applications in various fields, including manufacturing, engineering, and quality control. In the context of manufacturing, compound inequalities can be used to specify the tolerance range for various parts and components. This ensures that the final product meets the desired specifications and is of high quality.

Conclusion

In conclusion, the compound inequality −0.002≤x−3≤0.002-0.002 \leq x-3 \leq 0.002 represents the tolerance range for a manufactured part with a desired length of 3 mm. By solving this inequality, we can determine the range of values that the actual length of the part can take. This concept has numerous real-world applications in various fields, including manufacturing, engineering, and quality control.

Additional Examples

Example 1: Tolerance Range for a Part with a Desired Diameter

Suppose a quality assurance engineer is checking a part with a desired diameter of 5 mm. The tolerance range for the part is −0.005≤x−5≤0.005-0.005 \leq x-5 \leq 0.005. What is the range of values that the actual diameter of the part can take?

Solution

To solve the compound inequality, we need to isolate the variable x. We can do this by adding 5 to all three parts of the inequality:

−0.005+5≤x−5+5≤0.005+5-0.005 + 5 \leq x - 5 + 5 \leq 0.005 + 5

This simplifies to:

4.995≤x≤5.0054.995 \leq x \leq 5.005

Example 2: Tolerance Range for a Part with a Desired Thickness

Suppose a quality assurance engineer is checking a part with a desired thickness of 1.5 mm. The tolerance range for the part is −0.01≤x−1.5≤0.01-0.01 \leq x-1.5 \leq 0.01. What is the range of values that the actual thickness of the part can take?

Solution

To solve the compound inequality, we need to isolate the variable x. We can do this by adding 1.5 to all three parts of the inequality:

−0.01+1.5≤x−1.5+1.5≤0.01+1.5-0.01 + 1.5 \leq x - 1.5 + 1.5 \leq 0.01 + 1.5

This simplifies to:

1.49≤x≤1.511.49 \leq x \leq 1.51

Conclusion

Introduction

In our previous article, we explored the concept of compound inequalities and how they relate to quality assurance in manufacturing. We also provided examples of how compound inequalities can be used to specify the tolerance range for various parts and components. In this article, we will answer some frequently asked questions about compound inequalities and provide additional insights into their application in quality assurance.

Q: What is a compound inequality?

A: A compound inequality is a mathematical statement that combines two or more inequalities with a logical operator, such as "and" or "or". In the context of quality assurance, compound inequalities are used to specify the tolerance range for a part or component.

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to isolate the variable by adding or subtracting the same value to all three parts of the inequality. This will allow you to determine the range of values that the variable can take.

Q: What is the difference between a compound inequality and a single inequality?

A: A single inequality is a mathematical statement that compares a single value to a constant or another value. A compound inequality, on the other hand, combines two or more inequalities with a logical operator. In the context of quality assurance, compound inequalities are used to specify the tolerance range for a part or component, while single inequalities are used to specify a single value or a range of values.

Q: How do I determine the tolerance range for a part or component?

A: To determine the tolerance range for a part or component, you need to consider the desired specifications and the acceptable limits of variation. This can be done by using compound inequalities to specify the tolerance range.

Q: What are some common applications of compound inequalities in quality assurance?

A: Compound inequalities are used in various applications in quality assurance, including:

  • Specifying the tolerance range for parts and components
  • Determining the acceptable limits of variation for a process or product
  • Ensuring that products meet the desired specifications and are of high quality

Q: How do I use compound inequalities to specify the tolerance range for a part or component?

A: To use compound inequalities to specify the tolerance range for a part or component, you need to:

  1. Determine the desired specifications for the part or component
  2. Determine the acceptable limits of variation for the part or component
  3. Use compound inequalities to specify the tolerance range for the part or component

Q: What are some common mistakes to avoid when using compound inequalities in quality assurance?

A: Some common mistakes to avoid when using compound inequalities in quality assurance include:

  • Failing to consider the desired specifications and acceptable limits of variation
  • Using single inequalities instead of compound inequalities to specify the tolerance range
  • Failing to isolate the variable when solving the compound inequality

Conclusion

In conclusion, compound inequalities are a powerful tool in quality assurance, allowing us to specify the tolerance range for parts and components and ensure that products meet the desired specifications and are of high quality. By understanding and using compound inequalities correctly, quality assurance engineers can ensure that products are of high quality and meet the desired specifications.

Additional Resources