What Value Of $x$ Is In The Solution Set Of The Inequality $2(3x-1) \geq 4x-6$?A. − 10 -10 − 10 B. − 5 -5 − 5 C. − 3 -3 − 3 D. − 1 -1 − 1

Mar 12, 2025 by ADMIN 142 views

$What Value of $x$ is in the Solution Set of the Inequality $2(3x-1) \geq 4x-6$?$

Introduction

In mathematics, inequalities are a fundamental concept that deals with the comparison of two or more mathematical expressions. In this article, we will focus on solving a specific type of inequality, which is a linear inequality. We will explore the solution set of the inequality $2(3x-1) \geq 4x-6$ and determine the value of $x$ that satisfies this inequality.

Understanding the Inequality

The given inequality is $2(3x-1) \geq 4x-6$. To begin solving this inequality, we need to simplify the left-hand side by distributing the coefficient $2$ to the terms inside the parentheses. This gives us $6x-2 \geq 4x-6$.

Solving the Inequality

Now that we have simplified the inequality, we can proceed to solve for $x$. Our goal is to isolate the variable $x$ on one side of the inequality. To do this, we can start by subtracting $4x$ from both sides of the inequality. This gives us $2x-2 \geq -6$.

Isolating the Variable

Next, we need to isolate the variable $x$ by adding $2$ to both sides of the inequality. This gives us $2x \geq -4$.

Final Step

Finally, we can solve for $x$ by dividing both sides of the inequality by $2$. This gives us $x \geq -2$.

Conclusion

In conclusion, the solution set of the inequality $2(3x-1) \geq 4x-6$ is $x \geq -2$. This means that any value of $x$ that is greater than or equal to $-2$ satisfies the given inequality.

Checking the Answer Choices

Now that we have found the solution set of the inequality, we can check the answer choices to see which one satisfies the inequality. The answer choices are $-10$, $-5$, $-3$, and $-1$. We can plug each of these values into the inequality to see if it satisfies the inequality.

Checking Answer Choice A

Let's start by checking answer choice A, which is $-10$. Plugging this value into the inequality gives us $2(3(-10)-1) \geq 4(-10)-6$. Simplifying this expression gives us $-62 \geq -46$, which is true. Therefore, answer choice A satisfies the inequality.

Checking Answer Choice B

Next, let's check answer choice B, which is $-5$. Plugging this value into the inequality gives us $2(3(-5)-1) \geq 4(-5)-6$. Simplifying this expression gives us $-32 \geq -26$, which is true. Therefore, answer choice B satisfies the inequality.

Checking Answer Choice C

Now, let's check answer choice C, which is $-3$. Plugging this value into the inequality gives us $2(3(-3)-1) \geq 4(-3)-6$. Simplifying this expression gives us $-20 \geq -18$, which is true. Therefore, answer choice C satisfies the inequality.

Checking Answer Choice D

Finally, let's check answer choice D, which is $-1$. Plugging this value into the inequality gives us $2(3(-1)-1) \geq 4(-1)-6$. Simplifying this expression gives us $-8 \geq -10$, which is false. Therefore, answer choice D does not satisfy the inequality.

Conclusion

In conclusion, the value of $x$ that is in the solution set of the inequality $2(3x-1) \geq 4x-6$ is $-10$, $-5$, and $-3$. Therefore, the correct answer is A, B, and C.

Final Answer

The final answer is A, B, and C.

Introduction

In the previous article, we explored the solution set of the inequality $2(3x-1) \geq 4x-6$. We found that the solution set is $x \geq -2$. In this article, we will answer some frequently asked questions (FAQs) about solving linear inequalities.

Q: What is a linear inequality?

A: A linear inequality is an inequality that involves a linear expression. It is an inequality that can be written in the form $ax + b \geq cx + d$ or $ax + b \leq cx + d$, where $a$, $b$, $c$, and $d$ are constants.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable on one side of the inequality. You can do this by adding or subtracting the same value to both sides of the inequality, or by multiplying or dividing both sides of the inequality by the same non-zero value.

Q: What is the difference between a linear inequality and a linear equation?

A: A linear equation is an equation that involves a linear expression. It is an equation that can be written in the form $ax + b = cx + d$, where $a$, $b$, $c$, and $d$ are constants. A linear inequality, on the other hand, is an inequality that involves a linear expression.

Q: Can I use the same methods to solve a linear inequality as I would to solve a linear equation?

A: No, you cannot use the same methods to solve a linear inequality as you would to solve a linear equation. When solving a linear inequality, you need to consider the direction of the inequality, which is not the case when solving a linear equation.

Q: How do I determine the direction of the inequality?

A: To determine the direction of the inequality, 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 calculator to solve a linear inequality?

A: Yes, you can use a calculator to solve a linear inequality. However, you need to be careful when using a calculator to solve an inequality, as the calculator may not be able to handle the inequality correctly.

Q: How do I check my answer when solving a linear inequality?

A: To check your answer when solving a linear inequality, you need to plug your solution back into the original inequality and check if it is true. If it is true, then your solution is correct.

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

A: Some common mistakes to avoid when solving linear inequalities include:

Not considering the direction of the inequality
Not isolating the variable on one side of the inequality
Not checking the solution
Using the same methods to solve a linear inequality as you would to solve a linear equation

Conclusion

In conclusion, solving linear inequalities requires careful consideration of the direction of the inequality and the use of specific methods to isolate the variable. By following these methods and avoiding common mistakes, you can solve linear inequalities with confidence.

Final Answer

The final answer is that solving linear inequalities requires careful consideration of the direction of the inequality and the use of specific methods to isolate the variable.