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

Apr 2, 2025 by ADMIN 94 views

$What Value of $x$ is in the Solution Set of $3(x-4) \geq 5x+2$?$

Introduction

In mathematics, solving inequalities is a crucial concept that helps us understand the relationship between different variables. In this article, we will focus on solving the inequality $3(x-4) \geq 5x+2$ and finding the value of $x$ that is in the solution set. This involves using algebraic techniques to isolate the variable and determine the range of values that satisfy the inequality.

Understanding the Inequality

The given inequality is $3(x-4) \geq 5x+2$. To solve this, we need to isolate the variable $x$. The first step is to expand the left-hand side of the inequality using the distributive property. This gives us $3x-12 \geq 5x+2$.

Isolating the Variable

Next, we need to isolate the variable $x$. To do this, we can subtract $3x$ from both sides of the inequality, which gives us $-12 \geq 2x+2$. Then, we can subtract $2$ from both sides, resulting in $-14 \geq 2x$.

Solving for $x$

Now, we need to solve for $x$. To do this, we can divide both sides of the inequality by $2$, which gives us $-7 \geq x$. This means that the value of $x$ must be less than or equal to $-7$.

Checking the Solution

To check our solution, we can substitute $x=-7$ into the original inequality. This gives us $3(-7-4) \geq 5(-7)+2$, which simplifies to $-33 \geq -35$. Since this is true, we can conclude that $x=-7$ is in the solution set.

Conclusion

In conclusion, the value of $x$ that is in the solution set of $3(x-4) \geq 5x+2$ is $-7$. This means that any value of $x$ that is less than or equal to $-7$ will satisfy the inequality.

Step-by-Step Solution

Here is a step-by-step solution to the problem:

Expand the left-hand side of the inequality using the distributive property: $3(x-4) \geq 5x+2$ becomes $3x-12 \geq 5x+2$.
Subtract $3x$ from both sides of the inequality: $3x-12 \geq 5x+2$ becomes $-12 \geq 2x+2$.
Subtract $2$ from both sides of the inequality: $-12 \geq 2x+2$ becomes $-14 \geq 2x$.
Divide both sides of the inequality by $2$: $-14 \geq 2x$ becomes $-7 \geq x$.
Check the solution by substituting $x=-7$ into the original inequality: $3(-7-4) \geq 5(-7)+2$ becomes $-33 \geq -35$.

Frequently Asked Questions

What is the solution set of the inequality $3(x-4) \geq 5x+2$* How do I solve the inequality $3(x-4) \geq 5x+2$?
What is the value of $x$ that is in the solution set of the inequality $3(x-4) \geq 5x+2$?

Final Answer

The final answer is $\boxed{-7}$.

Introduction

Solving inequalities is a crucial concept in mathematics that helps us understand the relationship between different variables. In our previous article, we solved the inequality $3(x-4) \geq 5x+2$ and found the value of $x$ that is in the solution set. In this article, we will answer some frequently asked questions related to solving inequalities.

Q&A

Q: What is the solution set of the inequality $3(x-4) \geq 5x+2$?

A: The solution set of the inequality $3(x-4) \geq 5x+2$ is all values of $x$ that are less than or equal to $-7$.

Q: How do I solve the inequality $3(x-4) \geq 5x+2$?

A: To solve the inequality $3(x-4) \geq 5x+2$, you need to follow these steps:

Expand the left-hand side of the inequality using the distributive property.
Subtract $3x$ from both sides of the inequality.
Subtract $2$ from both sides of the inequality.
Divide both sides of the inequality by $2$.
Check the solution by substituting $x=-7$ into the original inequality.

Q: What is the value of $x$ that is in the solution set of the inequality $3(x-4) \geq 5x+2$?

A: The value of $x$ that is in the solution set of the inequality $3(x-4) \geq 5x+2$ is $-7$.

Q: How do I know if my solution is correct?

A: To check if your solution is correct, you need to substitute the value of $x$ into the original inequality and see if it is true. If it is true, then your solution is correct.

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

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

Not following the order of operations
Not isolating the variable correctly
Not checking the solution
Not considering the direction of the inequality

Q: How do I graph the solution set of an inequality?

A: To graph the solution set of an inequality, you need to follow these steps:

Draw a number line and mark the value of $x$ that is in the solution set.
Draw an open circle at the value of $x$ that is in the solution set.
Draw a closed circle at the value of $x$ that is not in the solution set.
Shade the region to the left of the closed circle if the inequality is less than or equal to.
Shade the region to the right of the closed circle if the inequality is greater than or equal to.