Break down the compound inequality into two separate inequalities

Oct 13, 2025 by windailys.com 7 views

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 these engineers 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 then use this scenario to represent a compound inequality and explore its meaning.

The Desired Specifications

Let's assume that the desired length of the part is 3 mm. This means that the quality assurance engineer wants to ensure that the actual length of the part is within a certain range of 3 mm. To be more precise, the engineer wants to check if the part is within 0.002 mm of the desired length of 3 mm.

The Compound Inequality

The compound inequality $-0.002 \leq x - 3 \leq 0.002$ represents the range of values that the actual length of the part can take. To understand this inequality, let's break it down into two separate inequalities:

$x - 3 \geq -0.002$
$x - 3 \leq 0.002$

Solving the Inequalities

Now, let's solve these two inequalities separately.

Inequality 1: $x - 3 \geq -0.002$

To solve this inequality, we need to isolate the variable x. We can do this by adding 3 to both sides of the inequality:

$x - 3 + 3 \geq -0.002 + 3$

This simplifies to:

$x \geq 2.998$

Inequality 2: $x - 3 \leq 0.002$

To solve this inequality, we need to isolate the variable x. We can do this by adding 3 to both sides of the inequality:

$x - 3 + 3 \leq 0.002 + 3$

This simplifies to:

$x \leq 3.002$

Combining the Inequalities

Now that we have solved the two inequalities, we can combine them to get the compound inequality:

$2.998 \leq x \leq 3.002$

This means that the actual length of the part must be between 2.998 mm and 3.002 mm, inclusive.

Conclusion

In this article, we used a scenario from the world of manufacturing to represent a compound inequality. We showed how to break down the compound inequality into two separate inequalities and solve them separately. We then combined the solutions to get the final compound inequality, which represents the range of values that the actual length of the part can take.

Applications of Compound Inequalities

Compound inequalities have many applications in real-world problems. Here are a few examples:

Quality control: As we saw in the scenario, compound inequalities can be used to represent the range of values that a manufactured part can take.
Finance: Compound inequalities can be used to represent the range of values that a stock price can take.
Science: Compound inequalities can be used to represent the range of values that a scientific measurement can take.

Tips for Solving Compound Inequalities

Here are a few tips for solving compound inequalities:

Break down the compound inequality into two separate inequalities: This will make it easier to solve the inequality.
Solve each inequality separately: This will help you to isolate the variable and find the solution.
Combine the solutions: Once you have solved each inequality, combine the solutions to get the final compound inequality.

Common Mistakes to Avoid

Here are a few common mistakes to avoid when solving compound inequalities:

Not breaking down the compound inequality into two separate inequalities: This can make it difficult to solve the inequality.
Not solving each inequality separately: This can lead to incorrect solutions.
Not combining the solutions: This can result in an incorrect final compound inequality.

Conclusion

In this article, we explored a scenario from the world of manufacturing to represent a compound inequality. We showed how to break down the compound inequality into two separate inequalities and solve them separately. We then combined the solutions to get the final compound inequality, which represents the range of values that the actual length of the part can take. We also discussed the applications of compound inequalities and provided tips for solving them. Finally, we highlighted common mistakes to avoid when solving compound inequalities.

Introduction

In our previous article, we explored a scenario from the world of manufacturing to represent a compound inequality. We showed how to break down the compound inequality into two separate inequalities and solve them separately. We then combined the solutions to get the final compound inequality, which represents the range of values that the actual length of the part can take. In this article, we will answer some frequently asked questions about compound inequalities.

Q&A

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".

Q: How do I solve a compound inequality?

A: To solve a compound inequality, you need to break it down into two separate inequalities and solve each one separately. Then, you combine the solutions to get the final compound inequality.

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 expression to a value, such as x > 3. A compound inequality, on the other hand, combines two or more inequalities with a logical operator.

Q: Can I use a compound inequality to represent a range of values?

A: Yes, a compound inequality can be used to represent a range of values. For example, the compound inequality 2 ≤ x ≤ 5 represents the range of values from 2 to 5, inclusive.

Q: How do I graph a compound inequality?

A: To graph a compound inequality, you need to graph each inequality separately and then combine the graphs. The resulting graph will show the range of values that satisfy the compound inequality.

Q: Can I use a compound inequality to represent a set of values?

A: Yes, a compound inequality can be used to represent a set of values. For example, the compound inequality {x | 2 ≤ x ≤ 5} represents the set of values from 2 to 5, inclusive.

Q: How do I determine the solution to a compound inequality?

A: To determine the solution to a compound inequality, you need to solve each inequality separately and then combine the solutions. The resulting solution will show the range of values that satisfy the compound inequality.

Q: Can I use a compound inequality to represent a real-world problem?

A: Yes, a compound inequality can be used to represent a real-world problem. For example, a quality assurance engineer might use a compound inequality to represent the range of values that a manufactured part can take.

Q: How do I check my solution to a compound inequality?

A: To check your solution to a compound inequality, you need to plug in values from the solution set into the original inequality and verify that they satisfy the inequality.

Q: Can I use a compound inequality to represent a system of inequalities?

A: Yes, a compound inequality can be used to represent a system of inequalities. For example, the compound inequality {x | 2 ≤ x ≤ 5 and 3 ≤ y ≤ 6} represents the system of inequalities 2 ≤ x ≤ 5 and 3 ≤ y ≤ 6.

Conclusion

In this article, we answered some frequently asked questions about compound inequalities. We discussed how to solve a compound inequality, how to graph a compound inequality, and how to determine the solution to a compound inequality. We also highlighted the importance of checking your solution to a compound inequality and using a compound inequality to represent a real-world problem.

Tips for Solving Compound Inequalities

Here are a few tips for solving compound inequalities:

Break down the compound inequality into two separate inequalities: This will make it easier to solve the inequality.
Solve each inequality separately: This will help you to isolate the variable and find the solution.
Combine the solutions: Once you have solved each inequality, combine the solutions to get the final compound inequality.
Check your solution: Plug in values from the solution set into the original inequality and verify that they satisfy the inequality.

Common Mistakes to Avoid

Here are a few common mistakes to avoid when solving compound inequalities:

Not breaking down the compound inequality into two separate inequalities: This can make it difficult to solve the inequality.
Not solving each inequality separately: This can lead to incorrect solutions.
Not combining the solutions: This can result in an incorrect final compound inequality.
Not checking your solution: This can lead to incorrect conclusions.