Compound Inequalities Solving Antonio's Commute Problem

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This is a classic problem that can be solved by understanding how to express real-world situations using mathematical inequalities. Let's break down the problem, explore the concepts involved, and arrive at the correct answer. We will delve into compound inequalities, focusing on the "or" case, and then apply this knowledge to Antonio's commuting situation. Finally, we will thoroughly analyze the given options to pinpoint the accurate representation of the problem.

Understanding Compound Inequalities

To effectively tackle this problem, it's crucial to have a solid grasp of compound inequalities. Compound inequalities are mathematical statements that combine two or more inequalities using the words "and" or "or." These inequalities help describe situations where a variable must satisfy multiple conditions simultaneously or satisfy at least one of several conditions.

When inequalities are joined by the word "and," it signifies that both conditions must be true at the same time. The solution set for an "and" compound inequality is the intersection of the solution sets of the individual inequalities. Graphically, this means the region where the solution sets overlap.

On the other hand, when inequalities are joined by the word "or," it means that at least one of the conditions must be true. The solution set for an "or" compound inequality is the union of the solution sets of the individual inequalities. Graphically, this means including all regions covered by either solution set.

In the context of real-world problems, compound inequalities with "or" often describe scenarios with alternative possibilities or conditions. For instance, a statement like "The temperature must be below freezing or above 80 degrees" is an example of an "or" compound inequality. The solution would include all temperatures that satisfy either condition.

Understanding the distinction between "and" and "or" is paramount when translating word problems into mathematical expressions. Recognizing which connective is appropriate is the key to accurately representing the given situation using inequalities.

Analyzing Antonio's Commute

Let's apply our understanding of compound inequalities to Antonio's commute. The problem states that Antonio wants to avoid heavy traffic. To do this, he has two options regarding his working hours:

  1. He can drive home after working less than 7 hours.
  2. He can drive home after working more than 9 hours.

Notice the crucial word here: "or." Antonio needs to satisfy either the first condition or the second condition to avoid traffic. This immediately tells us that we are dealing with an "or" compound inequality.

Now, let's represent these conditions mathematically. Let h represent the number of hours Antonio works. The first condition, "less than 7 hours," can be written as:

h < 7

The second condition, "more than 9 hours," can be written as:

h > 9

Since Antonio can avoid traffic if either of these conditions is met, we combine these inequalities using the word "or." This gives us the compound inequality:

h < 7 or h > 9

This compound inequality is the mathematical representation of the given scenario. It states that Antonio avoids heavy traffic if he works less than 7 hours or if he works more than 9 hours. This accurately captures the two possibilities presented in the problem statement.

The ability to translate verbal descriptions into mathematical expressions is fundamental in problem-solving. In this case, identifying the key phrase "less than" and "more than," as well as the connective "or," allows us to construct the compound inequality that models Antonio's commuting situation.

Evaluating the Options

Now, let's examine the options provided and see which one matches our derived compound inequality:

A. 7

This option is simply the number 7. It doesn't represent an inequality or a compound inequality, so it's incorrect.

B. 7>h>9

This option represents a compound inequality using "and." It states that 7 is greater than h, and h is greater than 9. This implies that h is simultaneously less than 7 and greater than 9, which is impossible. It also doesn't use the correct connective ("or") for the given scenario, so it's incorrect.

C. h<7 or h>9

This option perfectly matches our derived compound inequality. It states that h is less than 7 or h is greater than 9. This accurately represents the two conditions under which Antonio can avoid heavy traffic, so this is the correct answer.

D. h>7 or h<9

This option uses the correct connective ("or"), but the inequalities are reversed. It states that h is greater than 7 or h is less than 9. This inequality is true for almost all values of h (except h = 7). This doesn't accurately represent the given scenario, so it's incorrect.

By systematically analyzing each option and comparing it to our derived compound inequality, we can confidently identify the correct answer.

The Correct Answer

Based on our analysis, the correct answer is:

C. h<7 or h>9

This compound inequality accurately represents the conditions under which Antonio can avoid heavy traffic. It clearly states that he needs to work less than 7 hours or more than 9 hours.

Key Takeaways

This problem highlights the importance of understanding compound inequalities and their application in real-world scenarios. Key takeaways from this problem include:

  • Recognizing the difference between "and" and "or" in compound inequalities is crucial.
  • The word "or" indicates that at least one of the conditions must be true.
  • Translating verbal descriptions into mathematical expressions is a fundamental problem-solving skill.
  • Carefully analyzing each option and comparing it to the derived solution is essential to avoid errors.

By mastering these concepts, you can confidently tackle similar problems involving compound inequalities and real-world applications.

Compound inequalities can sometimes seem tricky, but with a clear understanding, we can easily solve them and apply them to real-world scenarios. Let’s take a look at a classic mathematical problem involving compound inequalities and its application to a commuting scenario. This article will dive deep into how to translate word problems into mathematical expressions, understand the difference between “and” and “or” in the context of inequalities, and accurately represent the given situation using a compound inequality.

Decoding Compound Inequalities: The Foundation

To effectively solve problems involving these situations, a strong understanding of compound inequalities is essential. Compound inequalities are mathematical statements that combine two or more inequalities using the words *