Solve: $2|x+7|-4 \geq 0$Express The Answer In Set-builder Notation.A. $\{x \mid 5\ \textless \ X\ \textless \ 6\}$ B. $\{x \mid X \geq-5\}$ C. $\{x \mid X \leq-9 \text{ Or } X \geq-5\}$ D. $\{x \mid-9\

Introduction

In this article, we will be solving the inequality $2|x+7|-4 \geq 0$ and expressing the answer in set-builder notation. Set-builder notation is a way of describing a set of numbers using a mathematical expression. It is commonly used in mathematics to describe the solution set of an inequality or an equation.

Understanding the Inequality

The given inequality is $2|x+7|-4 \geq 0$. This is an absolute value inequality, which means that the expression inside the absolute value bars can be either positive or negative. To solve this inequality, we need to consider both cases separately.

Case 1: $x+7 \geq 0$

In this case, the expression inside the absolute value bars is non-negative. We can rewrite the inequality as $2(x+7)-4 \geq 0$. Simplifying this expression, we get $2x+14-4 \geq 0$, which further simplifies to $2x+10 \geq 0$. Subtracting 10 from both sides, we get $2x \geq -10$. Dividing both sides by 2, we get $x \geq -5$.

Case 2: $x+7 < 0$

In this case, the expression inside the absolute value bars is negative. We can rewrite the inequality as $-2(x+7)-4 \geq 0$. Simplifying this expression, we get $-2x-14-4 \geq 0$, which further simplifies to $-2x-18 \geq 0$. Adding 18 to both sides, we get $-2x \geq -18$. Dividing both sides by -2, we get $x \leq -9$.

Combining the Cases

We have two cases: $x \geq -5$ and $x \leq -9$. However, these two cases are not mutually exclusive. In fact, they overlap, and the solution set is the union of the two cases.

Solution Set

The solution set of the inequality $2|x+7|-4 \geq 0$ is the set of all real numbers $x$ such that $x \leq -9$ or $x \geq -5$. This can be expressed in set-builder notation as $\{x \mid x \leq -9 \text{ or } x \geq -5\}$.

Conclusion

In this article, we solved the inequality $2|x+7|-4 \geq 0$ and expressed the answer in set-builder notation. We considered two cases separately and combined the solutions to find the final answer. The solution set is the union of the two cases, and it can be expressed in set-builder notation as $\{x \mid x \leq -9 \text{ or } x \geq -5\}$.

Comparison with Answer Choices

Let's compare our solution with the answer choices:

• A. $\{x \mid 5\ \textless \ x\ \textless \ 6\}$: This is not the correct solution, as it does not include the values $x \leq -9$ or $x \geq -5$.
• B. $\{x \mid x \geq-5\}$: This is not the correct solution, as it does not include the values $x \leq -9$.
• C. $\{x \mid x \leq-9 \text{ or } x \geq-5\}$: This is the correct solution, as it includes both the values $x \leq -9$ and $x \geq -5$.
• D. $\{x \mid-9\ \textless \ x\ \textless \ 6\}$: This is not the correct solution, as it does not include the values $x \geq -5$.

Therefore, the correct answer is C. $\{x \mid x \leq-9 \text{ or } x \geq-5\}$.

Final Answer

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

Introduction

In our previous article, we solved the inequality $2|x+7|-4 \geq 0$ and expressed the answer in set-builder notation. In this article, we will answer some frequently asked questions related to the solution of the inequality.

Q&A

Q1: What is the meaning of the absolute value inequality $2|x+7|-4 \geq 0$?

A1: The absolute value inequality $2|x+7|-4 \geq 0$ means that the expression $2|x+7|-4$ is greater than or equal to zero. In other words, the distance between $2|x+7|$ and 4 is non-negative.

Q2: How do we solve the absolute value inequality $2|x+7|-4 \geq 0$?

A2: To solve the absolute value inequality $2|x+7|-4 \geq 0$, we need to consider two cases separately: $x+7 \geq 0$ and $x+7 < 0$. We then simplify the inequality in each case and combine the solutions to find the final answer.

Q3: What is the solution set of the inequality $2|x+7|-4 \geq 0$?

A3: The solution set of the inequality $2|x+7|-4 \geq 0$ is the set of all real numbers $x$ such that $x \leq -9$ or $x \geq -5$. This can be expressed in set-builder notation as $\{x \mid x \leq -9 \text{ or } x \geq -5\}$.

Q4: How do we express the solution set in set-builder notation?

A4: To express the solution set in set-builder notation, we use the following format: $\{x \mid \text{condition}\}$. In this case, the condition is $x \leq -9$ or $x \geq -5$.

Q5: What is the difference between the solution set and the answer choices?

A5: The solution set is the set of all real numbers $x$ such that $x \leq -9$ or $x \geq -5$. The answer choices are specific sets of numbers that do not include the values $x \leq -9$ or $x \geq -5$.

Q6: Which answer choice is correct?

A6: The correct answer choice is C. $\{x \mid x \leq-9 \text{ or } x \geq-5\}$.

Conclusion

In this article, we answered some frequently asked questions related to the solution of the inequality $2|x+7|-4 \geq 0$. We explained the meaning of the absolute value inequality, how to solve it, and how to express the solution set in set-builder notation. We also compared the solution set with the answer choices and determined that the correct answer is C. $\{x \mid x \leq-9 \text{ or } x \geq-5\}$.

Final Answer

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