Solve The Following Inequality:$\[ -|x-3| \ \textgreater \ -4 \\]A. \[$(-1, 7)\$\]B. \[$(-\infty, -1] \cup [7, \infty)\$\]C. \[$[-1, 7]\$\]D. \[$(-\infty, -1) \cup (7, \infty)\$\]

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Introduction

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In this article, we will delve into the world of inequalities and solve a specific one step by step. The given inequality is $-|x-3| > -4$. We will break down the solution process into manageable parts, making it easy to understand and follow along. By the end of this article, you will have a clear understanding of how to solve this type of inequality and be able to apply the same steps to similar problems.

Understanding the Inequality

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Before we dive into the solution, let's take a closer look at the given inequality: $-|x-3| > -4$. The absolute value expression $|x-3|$ is the key to solving this inequality. We need to consider two cases: when $x-3$ is non-negative and when $x-3$ is negative.

Case 1: $x-3 \geq 0$

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When $x-3 \geq 0$, the absolute value expression $|x-3|$ simplifies to $x-3$. In this case, the inequality becomes:

$$-(x-3) > -4$$

To solve for $x$, we can start by multiplying both sides of the inequality by $-1$. However, when multiplying or dividing both sides of an inequality by a negative number, we need to reverse the direction of the inequality sign. Therefore, the inequality becomes:

$$x-3 < -4$$

Next, we can add $3$ to both sides of the inequality to isolate $x$:

$$x < -4 + 3$$

Simplifying the right-hand side, we get:

$$x < -1$$

Case 2: $x-3 < 0$

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When $x-3 < 0$, the absolute value expression $|x-3|$ simplifies to $-(x-3)$. In this case, the inequality becomes:

$$-(-(x-3)) > -4$$

Simplifying the left-hand side, we get:

$$x-3 > -4$$

Next, we can add $3$ to both sides of the inequality to isolate $x$:

$$x > -4 + 3$$

Simplifying the right-hand side, we get:

$$x > -1$$

Combining the Cases

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Now that we have solved the inequality for both cases, we need to combine the results. From Case 1, we have $x < -1$, and from Case 2, we have $x > -1$. However, we need to consider the original condition $x-3 \geq 0$ for Case 1 and $x-3 < 0$ for Case 2.

Case 1: $x-3 \geq 0$

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For Case 1, we have $x < -1$, and the condition $x-3 \geq 0$ is satisfied when $x \geq 3$. Therefore, the solution for Case 1 is:

$$x \in (-\infty, -1) \cup [3, \infty)$$

Case 2: $x-3 < 0$

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For Case 2, we have $x > -1$, and the condition $x-3 < 0$ is satisfied when $x < 3$. Therefore, the solution for Case 2 is:

$$x \in (-1, 3)$$

Final Solution

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Now that we have combined the results from both cases, we can write the final solution as:

$$x \in (-\infty, -1) \cup (3, \infty)$$

This solution represents the set of all values of $x$ that satisfy the original inequality $-|x-3| > -4$.

Conclusion

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In this article, we have solved the inequality $-|x-3| > -4$ step by step. We have considered two cases: when $x-3$ is non-negative and when $x-3$ is negative. By combining the results from both cases, we have arrived at the final solution:

$$x \in (-\infty, -1) \cup (3, \infty)$$

This solution represents the set of all values of $x$ that satisfy the original inequality. We hope that this article has provided a clear and concise explanation of how to solve this type of inequality and has helped you to understand the concept of absolute value inequalities.

Frequently Asked Questions

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Q: What is the absolute value inequality?

A: The absolute value inequality is an inequality that involves an absolute value expression. In this case, the absolute value expression is $|x-3|$.

Q: How do I solve an absolute value inequality?

A: To solve an absolute value inequality, you need to consider two cases: when the expression inside the absolute value is non-negative and when it is negative. You then need to solve the inequality for each case and combine the results.

Q: What is the final solution to the inequality $-|x-3| > -4$?

A: The final solution to the inequality $-|x-3| > -4$ is:

$$x \in (-\infty, -1) \cup (3, \infty)$$

This solution represents the set of all values of $x$ that satisfy the original inequality.

Q: Can I use a calculator to solve absolute value inequalities?

A: Yes, you can use a calculator to solve absolute value inequalities. However, it's always a good idea to understand the concept and solution process before relying on a calculator.

Q: How do I check my solution to an absolute value inequality?

A: To check your solution to an absolute value inequality, you can plug in values from the solution set into the original inequality and verify that it is true.

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Introduction

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In our previous article, we solved the inequality $-|x-3| > -4$ step by step. We also provided a comprehensive guide on how to solve absolute value inequalities. However, we know that there are many more questions and concerns that you may have. In this article, we will address some of the most frequently asked questions about absolute value inequalities.

Q&A

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Q: What is the difference between an absolute value inequality and a linear inequality?

A: An absolute value inequality is an inequality that involves an absolute value expression, whereas a linear inequality is an inequality that involves a linear expression. For example, the inequality $|x-3| > 2$ is an absolute value inequality, while the inequality $x > 2$ is a linear inequality.

Q: How do I know which case to use when solving an absolute value inequality?

A: When solving an absolute value inequality, you need to consider two cases: when the expression inside the absolute value is non-negative and when it is negative. To determine which case to use, you need to examine the inequality and determine whether the expression inside the absolute value is positive or negative.

Q: Can I use a calculator to solve absolute value inequalities?

A: Yes, you can use a calculator to solve absolute value inequalities. However, it's always a good idea to understand the concept and solution process before relying on a calculator.

Q: How do I check my solution to an absolute value inequality?

A: To check your solution to an absolute value inequality, you can plug in values from the solution set into the original inequality and verify that it is true.

Q: What is the final solution to the inequality $|x-3| > 2$?

A: The final solution to the inequality $|x-3| > 2$ is:

$$x \in (-\infty, 1) \cup (5, \infty)$$

This solution represents the set of all values of $x$ that satisfy the original inequality.

Q: Can I use the same steps to solve a compound inequality?

A: Yes, you can use the same steps to solve a compound inequality. However, you need to be careful when combining the results from each case.

Q: How do I solve an absolute value inequality with a fraction?

A: To solve an absolute value inequality with a fraction, you need to follow the same steps as before. However, you need to be careful when multiplying or dividing both sides of the inequality by a fraction.

Q: What is the final solution to the inequality $|x/2| > 3$?

A: The final solution to the inequality $|x/2| > 3$ is:

$$x \in (-\infty, -6) \cup (6, \infty)$$

This solution represents the set of all values of $x$ that satisfy the original inequality.

Conclusion

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In this article, we have addressed some of the most frequently asked questions about absolute value inequalities. We hope that this article has provided a clear and concise explanation of how to solve absolute value inequalities and has helped you to understand the concept and solution process.

Additional Resources

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If you have any further questions or concerns about absolute value inequalities, we recommend checking out the following resources:

• Khan Academy: Absolute Value Inequalities
• Mathway: Absolute Value Inequalities
• Wolfram Alpha: Absolute Value Inequalities

We hope that these resources will provide you with additional support and guidance as you learn about absolute value inequalities.

Final Thoughts

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Solving absolute value inequalities can be a challenging task, but with practice and patience, you can master the concept and solution process. Remember to always follow the steps outlined in this article and to check your solution to ensure that it is correct. With time and practice, you will become proficient in solving absolute value inequalities and will be able to tackle even the most complex problems with confidence.