What Is The Solution, If Any, To The Inequality $3 - |4 - N| \ \textgreater \ 1$?A. No Solution B. All Real Numbers C. $n \ \textgreater \ 2$ Or $ N \textless 6 N \ \textless \ 6 N \textless 6 [/tex] D. $2 \ \textless \ N \
Introduction
Inequalities involving absolute values can be challenging to solve, as they often require considering multiple cases based on the sign of the expression within the absolute value. In this article, we will explore the solution to the inequality $3 - |4 - n| \ \textgreater \ 1$, examining the different cases that arise and determining the set of values for which the inequality holds true.
Understanding Absolute Value Inequalities
Before diving into the specific inequality, it's essential to understand the general concept of absolute value inequalities. An absolute value inequality is of the form $|x| \ \textgreater \ a$, where $a$ is a real number. This inequality can be rewritten as two separate inequalities: $x \ \textgreater \ a$ and $x \ \textless \ -a$. When dealing with absolute value inequalities involving variables, we must consider the cases where the expression within the absolute value is positive or negative.
Case 1: $4 - n \ \textgreater \ 0$
In this case, the absolute value can be rewritten as $|4 - n| = 4 - n$. Substituting this into the original inequality, we get:
Simplifying the expression, we have:
Subtracting 1 from both sides, we obtain:
Case 2: $4 - n \ \textless \ 0$
In this case, the absolute value can be rewritten as $|4 - n| = -(4 - n) = n - 4$. Substituting this into the original inequality, we get:
Simplifying the expression, we have:
Subtracting 7 from both sides, we obtain:
Multiplying both sides by -1, we get:
Combining the Cases
We have found two separate cases, each with its own solution. However, we must consider the intersection of these cases to determine the overall solution to the inequality. The first case requires $n \ \textgreater \ 2$, while the second case requires $n \ \textless \ 6$. Combining these conditions, we find that the solution to the inequality is:
n \ \textgreater \ 2$ or $n \ \textless \ 6
Conclusion
In conclusion, the solution to the inequality $3 - |4 - n| \ \textgreater \ 1$ is $n \ \textgreater \ 2$ or $n \ \textless \ 6$. This solution arises from considering the two cases that arise when dealing with absolute value inequalities, and combining the conditions to determine the overall set of values for which the inequality holds true.
Final Answer
The final answer is: C. $n \ \textgreater \ 2$ or $n \ \textless \ 6$
Introduction
In the previous article, we explored the solution to the inequality $3 - |4 - n| \ \textgreater \ 1$. However, we understand that there may be many more questions and concerns about absolute value inequalities. In this article, we will address some of the most frequently asked questions (FAQs) about absolute value inequalities, providing clarity and insight into this complex topic.
Q: What is an absolute value inequality?
A: An absolute value inequality is an inequality that involves the absolute value of a variable or expression. It is of the form $|x| \ \textgreater \ a$, where $a$ is a real number.
Q: How do I solve an absolute value inequality?
A: To solve an absolute value inequality, you must consider two separate cases: one where the expression within the absolute value is positive, and one where it is negative. You can then rewrite the absolute value as a positive or negative expression, and solve the resulting inequality.
Q: What are the two cases for an absolute value inequality?
A: The two cases for an absolute value inequality are:
- Case 1: The expression within the absolute value is positive. In this case, the absolute value can be rewritten as $|x| = x$.
- Case 2: The expression within the absolute value is negative. In this case, the absolute value can be rewritten as $|x| = -x$.
Q: How do I determine which case to use?
A: To determine which case to use, you must consider the sign of the expression within the absolute value. If the expression is positive, use Case 1. If the expression is negative, use Case 2.
Q: What is the solution to the inequality $|x| \ \textgreater \ a$?
A: The solution to the inequality $|x| \ \textgreater \ a$ is $x \ \textgreater \ a$ or $x \ \textless \ -a$.
Q: Can I use the same method to solve absolute value inequalities with variables?
A: Yes, you can use the same method to solve absolute value inequalities with variables. However, you must consider the cases based on the sign of the expression within the absolute value, and then combine the conditions to determine the overall solution.
Q: What is the difference between an absolute value inequality and a linear inequality?
A: An absolute value inequality is an inequality that involves the absolute value of a variable or expression, while a linear inequality is an inequality that involves a linear expression. Absolute value inequalities can be more complex and require more careful consideration of the cases.
Q: Can I use algebraic manipulations to solve absolute value inequalities?
A: Yes, you can use algebraic manipulations to solve absolute value inequalities. However, you must be careful to consider the cases and combine the conditions correctly to determine the overall solution.
Q: What are some common mistakes to avoid when solving absolute value inequalities?
A: Some common mistakes to avoid when solving absolute value inequalities include:
- Failing to consider both cases
- Not combining the conditions correctly
- Not checking the solution for extraneous solutions
Conclusion
In conclusion, absolute value inequalities can be complex and challenging to solve. However, by understanding the two cases and combining the conditions correctly, you can determine the overall solution to the inequality. We hope that this FAQ article has provided clarity and insight into this topic, and has helped you to better understand absolute value inequalities.
Final Answer
The final answer is: There is no final numerical answer to this article, as it is a collection of FAQs about absolute value inequalities.