Twice The Sum Of A Number And 7 Is More Than 6 Or At Most -3. Which Compound Inequality Can Represent This Relationship?A. 2 ( X + 7 ) \textgreater 6 2(x+7)\ \textgreater \ 6 2 ( X + 7 ) \textgreater 6 And 2 ( X + 7 ) ≤ − 3 2(x+7) \leq-3 2 ( X + 7 ) ≤ − 3 B. 2 ( X + 7 ) \textless 6 2(x+7)\ \textless \ 6 2 ( X + 7 ) \textless 6 And $2(x+7)

Introduction

In mathematics, inequalities are used to represent relationships between different values or expressions. A compound inequality is a combination of two or more inequalities that are connected by logical operators such as "and" or "or". In this article, we will explore how to represent the relationship "twice the sum of a number and 7 is more than 6 or at most -3" using a compound inequality.

Understanding the Relationship

Let's break down the given relationship into its components:

• "Twice the sum of a number and 7" can be represented as 2(x + 7), where x is the number.
• "More than 6" can be represented as > 6.
• "At most -3" can be represented as ≤ -3.

Representing the Relationship as a Compound Inequality

To represent the relationship as a compound inequality, we need to combine the two inequalities using the "or" operator. The compound inequality can be written as:

2(x + 7) > 6 or 2(x + 7) ≤ -3

Analyzing the Options

Let's analyze the given options to determine which one represents the relationship:

A. $2(x+7)\ \textgreater \ 6$ and $2(x+7) \leq-3$

This option represents two separate inequalities that are connected by the "and" operator. However, the relationship we are trying to represent is "or", not "and".

B. $2(x+7)\ \textless \ 6$ and $2(x+7) \geq-3$

This option also represents two separate inequalities that are connected by the "and" operator. However, the relationship we are trying to represent is "or", not "and".

Conclusion

Based on our analysis, neither of the given options accurately represents the relationship "twice the sum of a number and 7 is more than 6 or at most -3". The correct representation of this relationship is:

2(x + 7) > 6 or 2(x + 7) ≤ -3

This compound inequality accurately represents the relationship between the sum of a number and 7, and the values 6 and -3.

Tips for Solving Compound Inequalities

When solving compound inequalities, it's essential to remember the following tips:

• Use the "or" operator to combine the inequalities.
• Make sure to include both inequalities in the compound inequality.
• Use the correct logical operator (e.g., "and" or "or") to connect the inequalities.

Examples of Compound Inequalities

Here are a few examples of compound inequalities:

• 2x + 5 > 10 or 2x + 5 ≤ 3
• x - 2 > 4 or x - 2 ≤ -1
• 3x + 1 > 7 or 3x + 1 ≤ -2

Conclusion

In conclusion, compound inequalities are a powerful tool for representing relationships between different values or expressions. By understanding how to represent relationships using compound inequalities, we can solve a wide range of mathematical problems. Remember to use the "or" operator to combine the inequalities, and make sure to include both inequalities in the compound inequality. With practice and patience, you'll become proficient in solving compound inequalities and tackling complex mathematical problems.