Which Are Correct Representations Of The Inequality 6 X ≥ 3 + 4 ( 2 X − 1 6x \geq 3 + 4(2x - 1 6 X ≥ 3 + 4 ( 2 X − 1 ]? Select Three Options.A. 1 ≥ 2 X 1 \geq 2x 1 ≥ 2 X B. 6 X ≥ 3 + 8 X − 4 6x \geq 3 + 8x - 4 6 X ≥ 3 + 8 X − 4 (Note: More Options Are Needed For A Complete Multiple-choice Question, As Only Two Are

Understanding the Basics of Inequalities

Inequalities are mathematical expressions that compare two values, indicating whether one is greater than, less than, or equal to the other. In this article, we will focus on solving inequalities, specifically the inequality $6x \geq 3 + 4(2x - 1)$.

The Importance of Following the Order of Operations

When solving inequalities, it is crucial to follow the order of operations (PEMDAS). This means that we need to evaluate expressions inside parentheses first, followed by exponents, multiplication and division, and finally addition and subtraction.

Solving the Inequality $6x \geq 3 + 4(2x - 1)$

To solve the inequality $6x \geq 3 + 4(2x - 1)$, we need to follow the order of operations.

Step 1: Evaluate the Expression Inside the Parentheses

First, we need to evaluate the expression inside the parentheses: $4(2x - 1)$. This can be rewritten as $8x - 4$.

Step 2: Rewrite the Inequality

Now that we have evaluated the expression inside the parentheses, we can rewrite the inequality as $6x \geq 3 + 8x - 4$.

Step 3: Combine Like Terms

Next, we need to combine like terms on the right-hand side of the inequality. This means that we need to combine the constants $3$ and $-4$.

Step 4: Simplify the Inequality

After combining like terms, we can simplify the inequality to $6x \geq 8x - 1$.

Step 5: Isolate the Variable

To isolate the variable $x$, we need to get all the terms with $x$ on one side of the inequality. We can do this by subtracting $8x$ from both sides of the inequality.

Step 6: Solve for $x$

After isolating the variable, we can solve for $x$ by dividing both sides of the inequality by $-2$.

The Correct Representations of the Inequality

Now that we have solved the inequality, we can determine the correct representations of the inequality.

Option A: $1 \geq 2x$

To determine if option A is correct, we need to check if it is equivalent to the original inequality. We can do this by dividing both sides of the inequality by $2$.

Option B: $6x \geq 3 + 8x - 4$

To determine if option B is correct, we need to check if it is equivalent to the original inequality. We can do this by combining like terms on the right-hand side of the inequality.

Option C: $-2x \geq -1$

To determine if option C is correct, we need to check if it is equivalent to the original inequality. We can do this by dividing both sides of the inequality by $-2$.

Option D: $x \leq \frac{1}{2}$

To determine if option D is correct, we need to check if it is equivalent to the original inequality. We can do this by dividing both sides of the inequality by $2$.

Conclusion

In conclusion, the correct representations of the inequality $6x \geq 3 + 4(2x - 1)$ are:

• Option A: $1 \geq 2x$
• Option B: $6x \geq 3 + 8x - 4$
• Option C: $-2x \geq -1$
• Option D: $x \leq \frac{1}{2}$

These options are all equivalent to the original inequality and can be used as correct representations of the inequality.

Understanding Inequalities

Inequalities are mathematical expressions that compare two values, indicating whether one is greater than, less than, or equal to the other. In this article, we will focus on solving inequalities, specifically the inequality $6x \geq 3 + 4(2x - 1)$.

Q&A: Solving Inequalities

Q: What is the first step in solving an inequality?

A: The first step in solving an inequality is to evaluate the expression inside any parentheses. This means that we need to follow the order of operations (PEMDAS).

Q: How do I evaluate the expression inside the parentheses?

A: To evaluate the expression inside the parentheses, we need to follow the order of operations (PEMDAS). This means that we need to multiply the numbers inside the parentheses first, followed by any exponents, and finally any addition or subtraction.

Q: What is the next step in solving an inequality?

A: After evaluating the expression inside the parentheses, the next step is to rewrite the inequality with the evaluated expression.

Q: How do I combine like terms on the right-hand side of the inequality?

A: To combine like terms on the right-hand side of the inequality, we need to add or subtract the coefficients of the like terms.

Q: What is the final step in solving an inequality?

A: The final step in solving an inequality is to isolate the variable by getting all the terms with the variable on one side of the inequality.

Q: How do I determine if an option is correct?

A: To determine if an option is correct, we need to check if it is equivalent to the original inequality. We can do this by following the steps outlined above.

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 (PEMDAS)
• Not evaluating the expression inside the parentheses
• Not combining like terms on the right-hand side of the inequality
• Not isolating the variable

Q: How do I check if an option is correct?

A: To check if an option is correct, we need to follow the steps outlined above and make sure that the option is equivalent to the original inequality.

Q: What are some tips for solving inequalities?

A: Some tips for solving inequalities include:

• Following the order of operations (PEMDAS)
• Evaluating the expression inside the parentheses
• Combining like terms on the right-hand side of the inequality
• Isolating the variable
• Checking the option against the original inequality

Conclusion

In conclusion, solving inequalities requires following the order of operations (PEMDAS), evaluating the expression inside the parentheses, combining like terms on the right-hand side of the inequality, and isolating the variable. By following these steps and avoiding common mistakes, we can determine if an option is correct and solve inequalities with confidence.

Additional Resources

For more information on solving inequalities, please refer to the following resources:

• Mathway
• Khan Academy
• Math Open Reference

Final Thoughts

Solving inequalities can be a challenging task, but with practice and patience, we can become proficient in solving them. By following the steps outlined above and avoiding common mistakes, we can determine if an option is correct and solve inequalities with confidence.