Which Number Line Represents The Solution Set For The Inequality 3 ( 8 − 4 X ) \textless 6 ( X − 5 3(8-4x) \ \textless \ 6(x-5 3 ( 8 − 4 X ) \textless 6 ( X − 5 ]?

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Introduction

In mathematics, inequalities are used to compare two expressions and determine the relationship between them. The inequality $3(8-4x) \ \textless \ 6(x-5)$ is a linear inequality that involves two variables, $x$ and the constant term. In this article, we will explore the solution set for this inequality and determine which number line represents the solution set.

Understanding the Inequality

The given inequality is $3(8-4x) \ \textless \ 6(x-5)$. To begin solving this inequality, we need to simplify the expressions on both sides. We can start by distributing the numbers outside the parentheses to the terms inside.

3(8-4x) = 24 - 12x
6(x-5) = 6x - 30

Now, we can rewrite the inequality as:

24 - 12x \  \textless \  6x - 30

Solving the Inequality

To solve the inequality, we need to isolate the variable $x$ on one side of the inequality. We can start by adding $12x$ to both sides of the inequality to get:

24 \  \textless \  18x - 30

Next, we can add $30$ to both sides of the inequality to get:

54 \  \textless \  18x

Now, we can divide both sides of the inequality by $18$ to get:

x \  \textless \  3

Graphing the Solution Set

The solution set for the inequality $x \ \textless \ 3$ is all real numbers less than $3$. We can represent this solution set on a number line by drawing a line at $x=3$ and shading the region to the left of the line.

x \  \textless \  3

Conclusion

In this article, we explored the solution set for the inequality $3(8-4x) \ \textless \ 6(x-5)$. We simplified the expressions on both sides of the inequality and isolated the variable $x$ on one side. We then graphed the solution set on a number line and determined that the solution set is all real numbers less than $3$.

Frequently Asked Questions

• What is the solution set for the inequality $3(8-4x) \ \textless \ 6(x-5)$?
• How do we simplify the expressions on both sides of the inequality?
• How do we isolate the variable $x$ on one side of the inequality?
• What is the graph of the solution set on a number line?

Step-by-Step Solution

1. Simplify the expressions on both sides of the inequality.
2. Add $12x$ to both sides of the inequality.
3. Add $30$ to both sides of the inequality.
4. Divide both sides of the inequality by $18$.
5. Graph the solution set on a number line.

Common Mistakes

• Failing to simplify the expressions on both sides of the inequality.
• Failing to isolate the variable $x$ on one side of the inequality.
• Graphing the solution set incorrectly on a number line.

Real-World Applications

• In finance, inequalities are used to compare the value of investments and determine the relationship between them.
• In science, inequalities are used to compare the values of physical quantities and determine the relationship between them.
• In engineering, inequalities are used to compare the values of design parameters and determine the relationship between them.

Conclusion

In conclusion, the solution set for the inequality $3(8-4x) \ \textless \ 6(x-5)$ is all real numbers less than $3$. We simplified the expressions on both sides of the inequality and isolated the variable $x$ on one side. We then graphed the solution set on a number line and determined that the solution set is all real numbers less than $3$. This article provides a step-by-step solution to the inequality and highlights common mistakes to avoid.

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Introduction

In the previous article, we explored the solution set for the inequality $3(8-4x) \ \textless \ 6(x-5)$. We simplified the expressions on both sides of the inequality, isolated the variable $x$ on one side, and graphed the solution set on a number line. In this article, we will answer some frequently asked questions (FAQs) related to the inequality.

Q&A

Q1: What is the solution set for the inequality $3(8-4x) \ \textless \ 6(x-5)$?

A1: The solution set for the inequality $3(8-4x) \ \textless \ 6(x-5)$ is all real numbers less than $3$.

Q2: How do we simplify the expressions on both sides of the inequality?

A2: To simplify the expressions on both sides of the inequality, we need to distribute the numbers outside the parentheses to the terms inside. We can start by distributing $3$ to the terms inside the parentheses on the left side and $6$ to the terms inside the parentheses on the right side.

Q3: How do we isolate the variable $x$ on one side of the inequality?

A3: To isolate the variable $x$ on one side of the inequality, we need to add or subtract the same value from both sides of the inequality. In this case, we added $12x$ to both sides of the inequality to get $24 \ \textless \ 18x - 30$. We then added $30$ to both sides of the inequality to get $54 \ \textless \ 18x$. Finally, we divided both sides of the inequality by $18$ to get $x \ \textless \ 3$.

Q4: What is the graph of the solution set on a number line?

A4: The graph of the solution set on a number line is a line at $x=3$ with the region to the left of the line shaded. This represents all real numbers less than $3$.

Q5: Can we use the same method to solve other inequalities?

A5: Yes, we can use the same method to solve other inequalities. The steps involved in solving an inequality are similar to those involved in solving an equation. We need to simplify the expressions on both sides of the inequality, isolate the variable on one side, and graph the solution set on a number line.

Q6: What are some common mistakes to avoid when solving inequalities?

A6: Some common mistakes to avoid when solving inequalities include failing to simplify the expressions on both sides of the inequality, failing to isolate the variable on one side, and graphing the solution set incorrectly on a number line.

Q7: How do we determine the direction of the inequality?

A7: To determine the direction of the inequality, we need to consider the signs of the coefficients of the variable. If the coefficient of the variable is positive, the inequality is greater than or equal to. If the coefficient of the variable is negative, the inequality is less than or equal to.

Q8: Can we use inequalities to solve real-world problems?

A8: Yes, we can use inequalities to solve real-world problems. Inequalities are used to compare the values of physical quantities and determine the relationship between them. They are also used to compare the values of design parameters and determine the relationship between them.

Conclusion

In conclusion, the FAQs for the inequality $3(8-4x) \ \textless \ 6(x-5)$ provide a comprehensive overview of the solution set, the steps involved in solving the inequality, and some common mistakes to avoid. We hope that this article has been helpful in answering some of the frequently asked questions related to the inequality.

Additional Resources

• For more information on inequalities, please refer to the article "Understanding Inequalities".
• For more information on graphing inequalities, please refer to the article "Graphing Inequalities".
• For more information on solving real-world problems using inequalities, please refer to the article "Solving Real-World Problems Using Inequalities".

Related Articles

• "Understanding Inequalities"
• "Graphing Inequalities"
• "Solving Real-World Problems Using Inequalities"

Tags

• Inequalities
• Graphing Inequalities
• Solving Real-World Problems Using Inequalities
• FAQs
• Solution Set
• Graphing
• Inequality