What Value Of $x$ Is In The Solution Set Of The Inequality − 2 ( 3 X + 2 ) \textgreater − 8 X + 6 -2(3x+2) \ \textgreater \ -8x+6 − 2 ( 3 X + 2 ) \textgreater − 8 X + 6 ?A. − 6 -6 − 6 B. − 5 -5 − 5 C. 5 5 5 D. 6 6 6

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Introduction

In mathematics, inequalities are a fundamental concept that help us compare the values of different expressions. Solving inequalities involves finding the values of the variable that satisfy the given inequality. In this article, we will focus on solving the inequality $-2(3x+2) \ \textgreater \ -8x+6$ and finding the value of $x$ that is in the solution set.

Understanding the Inequality

The given inequality is $-2(3x+2) \ \textgreater \ -8x+6$. To solve this inequality, we need to follow the order of operations (PEMDAS) and simplify the expressions. First, we will distribute the negative 2 to the terms inside the parentheses:

$$-2(3x+2) = -6x - 4$$

Now, the inequality becomes:

$$-6x - 4 \ \textgreater \ -8x+6$$

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 $6x$ to both sides of the inequality:

$$-6x - 4 + 6x \ \textgreater \ -8x+6 + 6x$$

This simplifies to:

$$-4 \ \textgreater \ -2x+6$$

Next, we can subtract 6 from both sides of the inequality:

$$-4 - 6 \ \textgreater \ -2x+6 - 6$$

This simplifies to:

$$-10 \ \textgreater \ -2x$$

Isolating the Variable

To isolate the variable $x$, we need to get rid of the negative sign in front of the $-2x$. We can do this by multiplying both sides of the inequality by $-1$:

$$-1(-10) \ \textless \ -1(-2x)$$

This simplifies to:

$$10 \ \textless \ 2x$$

Final Step

To isolate the variable $x$, we need to get rid of the 2 that is being multiplied to $x$. We can do this by dividing both sides of the inequality by 2:

$$\frac{10}{2} \ \textless \ \frac{2x}{2}$$

This simplifies to:

$$5 \ \textless \ x$$

Conclusion

The solution set of the inequality $-2(3x+2) \ \textgreater \ -8x+6$ is all values of $x$ that are greater than 5. Therefore, the value of $x$ that is in the solution set is $\boxed{5}$.

Final Answer

The final answer is $\boxed{5}$.

Discussion

The solution to this inequality is a simple example of how to solve linear inequalities. However, it is essential to remember that when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. This is a crucial step in solving inequalities and must be done carefully to avoid errors.

Additional Tips

When solving inequalities, it is essential to follow the order of operations (PEMDAS) and simplify the expressions before isolating the variable. Additionally, when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed.

Common Mistakes

One common mistake when solving inequalities is to forget to reverse the direction of the inequality sign when multiplying or dividing both sides by a negative number. This can lead to incorrect solutions and must be avoided.

Real-World Applications

Inequalities have many real-world applications, such as solving problems involving rates, ratios, and proportions. They are also used in finance, economics, and engineering to model and solve problems.

Conclusion

In conclusion, solving the inequality $-2(3x+2) \ \textgreater \ -8x+6$ involves following the order of operations (PEMDAS), simplifying the expressions, and isolating the variable. The solution set of the inequality is all values of $x$ that are greater than 5. Therefore, the value of $x$ that is in the solution set is $\boxed{5}$.

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Introduction

In our previous article, we solved the inequality $-2(3x+2) \ \textgreater \ -8x+6$ and found that the solution set is all values of $x$ that are greater than 5. In this article, we will answer some frequently asked questions about solving this inequality.

Q: What is the first step in solving the inequality $-2(3x+2) \ \textgreater \ -8x+6$?

A: The first step in solving the inequality is to distribute the negative 2 to the terms inside the parentheses. This gives us $-6x - 4 \ \textgreater \ -8x+6$.

Q: Why do we need to follow the order of operations (PEMDAS) when solving inequalities?

A: Following the order of operations (PEMDAS) is essential when solving inequalities because it ensures that we simplify the expressions correctly before isolating the variable. If we don't follow the order of operations, we may end up with an incorrect solution.

Q: What happens when we multiply or divide both sides of an inequality by a negative number?

A: When we multiply or divide both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. For example, if we have the inequality $a \ \textgreater \ b$ and we multiply both sides by $-1$, the inequality becomes $-a \ \textless \ -b$.

Q: How do we isolate the variable $x$ in the inequality $-2(3x+2) \ \textgreater \ -8x+6$?

A: To isolate the variable $x$, we need to get rid of the negative sign in front of the $-2x$. We can do this by multiplying both sides of the inequality by $-1$. This gives us $10 \ \textless \ 2x$. Then, we can divide both sides of the inequality by 2 to get $5 \ \textless \ x$.

Q: What is the solution set of the inequality $-2(3x+2) \ \textgreater \ -8x+6$?

A: The solution set of the inequality is all values of $x$ that are greater than 5.

Q: Why is it essential to check our solutions when solving inequalities?

A: It is essential to check our solutions when solving inequalities because we may have made a mistake in our calculations. By checking our solutions, we can ensure that we have found the correct solution set.

Q: How do we check our solutions when solving inequalities?

A: To check our solutions, we can substitute the solution into the original inequality and see if it is true. If the solution satisfies the inequality, then it is part of the solution set.

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

A: Some common mistakes to avoid when solving inequalities include forgetting to reverse the direction of the inequality sign when multiplying or dividing both sides by a negative number, and not following the order of operations (PEMDAS).

Q: How do we apply the concept of inequalities in real-world problems?

A: The concept of inequalities is applied in many real-world problems, such as solving problems involving rates, ratios, and proportions. Inequalities are also used in finance, economics, and engineering to model and solve problems.

Conclusion

In conclusion, solving the inequality $-2(3x+2) \ \textgreater \ -8x+6$ involves following the order of operations (PEMDAS), simplifying the expressions, and isolating the variable. The solution set of the inequality is all values of $x$ that are greater than 5. By following the steps outlined in this article, you can solve inequalities and apply the concept of inequalities in real-world problems.

Final Answer

The final answer is $\boxed{5}$.

Discussion

The solution to this inequality is a simple example of how to solve linear inequalities. However, it is essential to remember that when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed. This is a crucial step in solving inequalities and must be done carefully to avoid errors.

Additional Tips

When solving inequalities, it is essential to follow the order of operations (PEMDAS) and simplify the expressions before isolating the variable. Additionally, when multiplying or dividing both sides of an inequality by a negative number, the direction of the inequality sign must be reversed.

Common Mistakes

One common mistake when solving inequalities is to forget to reverse the direction of the inequality sign when multiplying or dividing both sides by a negative number. This can lead to incorrect solutions and must be avoided.

Real-World Applications

Inequalities have many real-world applications, such as solving problems involving rates, ratios, and proportions. They are also used in finance, economics, and engineering to model and solve problems.

Conclusion

In conclusion, solving the inequality $-2(3x+2) \ \textgreater \ -8x+6$ involves following the order of operations (PEMDAS), simplifying the expressions, and isolating the variable. The solution set of the inequality is all values of $x$ that are greater than 5. By following the steps outlined in this article, you can solve inequalities and apply the concept of inequalities in real-world problems.