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

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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 terms. To solve this inequality, we need to isolate the variable x and determine the solution set. In this article, we will explore the steps to solve the inequality and represent the solution set on a number line.

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.

Distributing the Numbers

When we distribute the numbers outside the parentheses to the terms inside, we get:

$3(8 - 4x) = 24 - 12x$

$6(x - 5) = 6x - 30$

Now, the inequality becomes:

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

Combining Like Terms

To simplify the inequality further, we can combine like terms. We can add 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:

$54 \ \textless \ 18x$

Isolating the Variable

Now, we can isolate the variable x by dividing both sides of the inequality by 18. This gives us:

$3 \ \textless \ x$

Representing the Solution Set on a Number Line

The solution set for the inequality $3 \ \textless \ x$ represents all the values of x that satisfy the inequality. To represent the solution set on a number line, we can draw a line and mark the point x = 3. Since the inequality is strict, we need to shade the region to the right of the point x = 3.

Number Line Representation

Here is the number line representation of the solution set:

• The point x = 3 is marked on the number line.
• The region to the right of the point x = 3 is shaded.

Conclusion

In this article, we solved the inequality $3(8 - 4x) \ \textless \ 6(x - 5)$ and represented the solution set on a number line. We simplified the expressions on both sides of the inequality, combined like terms, isolated the variable x, and represented the solution set on a number line. The solution set for the inequality $3 \ \textless \ x$ represents all the values of x that satisfy the inequality, and it is represented by the number line with the point x = 3 marked and the region to the right of the point x = 3 shaded.

Frequently Asked Questions

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

A: The solution set for the inequality $3(8 - 4x) \ \textless \ 6(x - 5)$ is x > 3.

Q: How do I represent the solution set on a number line?

A: To represent the solution set on a number line, you can draw a line and mark the point x = 3. Since the inequality is strict, you need to shade the region to the right of the point x = 3.

Q: What is the point x = 3 on the number line?

A: The point x = 3 on the number line represents the value of x that satisfies the inequality $3 \ \textless \ x$.

Q: What is the region to the right of the point x = 3 on the number line?

A: The region to the right of the point x = 3 on the number line represents all the values of x that satisfy the inequality $3 \ \textless \ x$.

Step-by-Step Solution

Step 1: Simplify the Expressions

Simplify the expressions on both sides of the inequality by distributing the numbers outside the parentheses to the terms inside.

$3(8 - 4x) = 24 - 12x$

$6(x - 5) = 6x - 30$

Step 2: Combine Like Terms

Combine like terms by adding 12x to both sides of the inequality to get:

$24 \ \textless \ 18x - 30$

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

$54 \ \textless \ 18x$

Step 3: Isolate the Variable

Isolate the variable x by dividing both sides of the inequality by 18. This gives us:

$3 \ \textless \ x$

Step 4: Represent the Solution Set on a Number Line

Represent the solution set on a number line by drawing a line and marking the point x = 3. Since the inequality is strict, shade the region to the right of the point x = 3.

Final Answer

The final answer is x > 3.

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Introduction

In the previous article, we solved the inequality $3(8 - 4x) \ \textless \ 6(x - 5)$ and represented the solution set on a number line. In this article, we will answer some frequently asked questions (FAQs) about the inequality.

Q&A

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

A: The solution set for the inequality $3(8 - 4x) \ \textless \ 6(x - 5)$ is x > 3.

Q: How do I represent the solution set on a number line?

A: To represent the solution set on a number line, you can draw a line and mark the point x = 3. Since the inequality is strict, you need to shade the region to the right of the point x = 3.

Q: What is the point x = 3 on the number line?

A: The point x = 3 on the number line represents the value of x that satisfies the inequality $3 \ \textless \ x$.

Q: What is the region to the right of the point x = 3 on the number line?

A: The region to the right of the point x = 3 on the number line represents all the values of x that satisfy the inequality $3 \ \textless \ x$.

Q: Can I use a number line to represent the solution set for a linear inequality with a fraction?

A: Yes, you can use a number line to represent the solution set for a linear inequality with a fraction. However, you need to be careful when working with fractions, as they can lead to more complex inequalities.

Q: How do I determine the direction of the inequality on a number line?

A: To determine the direction of the inequality on a number line, you need to consider the sign of the coefficient of the variable. If the coefficient is positive, the inequality is greater than or equal to. If the coefficient is negative, the inequality is less than or equal to.

Q: Can I use a number line to represent the solution set for a linear inequality with absolute value?

A: Yes, you can use a number line to represent the solution set for a linear inequality with absolute value. However, you need to be careful when working with absolute value, as it can lead to more complex inequalities.

Q: How do I represent the solution set for a linear inequality with multiple variables?

A: To represent the solution set for a linear inequality with multiple variables, you need to use a three-dimensional number line or a graph. This can be more complex than representing the solution set for a linear inequality with a single variable.

Conclusion

In this article, we answered some frequently asked questions (FAQs) about the inequality $3(8 - 4x) \ \textless \ 6(x - 5)$. We covered topics such as representing the solution set on a number line, determining the direction of the inequality, and working with fractions and absolute value.

Final Answer

The final answer is x > 3.

Step-by-Step Solution

Step 1: Simplify the Expressions

Simplify the expressions on both sides of the inequality by distributing the numbers outside the parentheses to the terms inside.

$3(8 - 4x) = 24 - 12x$

$6(x - 5) = 6x - 30$

Step 2: Combine Like Terms

Combine like terms by adding 12x to both sides of the inequality to get:

$24 \ \textless \ 18x - 30$

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

$54 \ \textless \ 18x$

Step 3: Isolate the Variable

Isolate the variable x by dividing both sides of the inequality by 18. This gives us:

$3 \ \textless \ x$

Step 4: Represent the Solution Set on a Number Line

Represent the solution set on a number line by drawing a line and marking the point x = 3. Since the inequality is strict, shade the region to the right of the point x = 3.

Final Answer

The final answer is x > 3.