What Is The Solution, If Any, To The Inequality 3 − ∣ 4 − 7 ∣ \textgreater 1 3-|4-7|\ \textgreater \ 1 3 − ∣4 − 7∣ \textgreater 1 ?A. No SolutionB. All Real NumbersC. N \textgreater 2 N\ \textgreater \ 2 N \textgreater 2 Or N \textless 6 N\ \textless \ 6 N \textless 6 D. 2 \textless N \textless 6 2\ \textless \ N\ \textless \ 6 2 \textless N \textless 6

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Understanding the Inequality

The given inequality is $3-|4-7|\ \textgreater \ 1$. To solve this inequality, we need to follow the order of operations (PEMDAS) and simplify the expression inside the absolute value first.

Simplifying the Expression Inside the Absolute Value

The expression inside the absolute value is $4-7$, which simplifies to $-3$. Therefore, the inequality becomes $3-|(-3)|\ \textgreater \ 1$.

Evaluating the Absolute Value

The absolute value of $-3$ is $3$, so the inequality becomes $3-3\ \textgreater \ 1$.

Simplifying the Inequality

Simplifying the inequality, we get $0\ \textgreater \ 1$.

Conclusion

Since $0$ is not greater than $1$, the inequality has no solution.

Solution Explanation

The correct answer is A. No solution. This is because the inequality $0\ \textgreater \ 1$ is never true, and therefore, there is no value of $n$ that satisfies the inequality.

Alternative Solution

Let's consider an alternative approach to solving the inequality. We can start by evaluating the expression inside the absolute value, which is $4-7$. This simplifies to $-3$. Therefore, the inequality becomes $3-|(-3)|\ \textgreater \ 1$.

Evaluating the Absolute Value

The absolute value of $-3$ is $3$, so the inequality becomes $3-3\ \textgreater \ 1$.

Simplifying the Inequality

Simplifying the inequality, we get $0\ \textgreater \ 1$.

Conclusion

Since $0$ is not greater than $1$, the inequality has no solution.

Solution Explanation

The correct answer is A. No solution. This is because the inequality $0\ \textgreater \ 1$ is never true, and therefore, there is no value of $n$ that satisfies the inequality.

Comparison with Other Options

Let's compare the solution to the inequality with the other options.

Option B: All Real Numbers

This option is incorrect because the inequality $0\ \textgreater \ 1$ is never true, and therefore, there is no value of $n$ that satisfies the inequality.

Option C: $n\ \textgreater \ 2$ or $n\ \textless \ 6$

This option is incorrect because the inequality $0\ \textgreater \ 1$ is never true, and therefore, there is no value of $n$ that satisfies the inequality.

Option D: $2\ \textless \ n\ \textless \ 6$

This option is incorrect because the inequality $0\ \textgreater \ 1$ is never true, and therefore, there is no value of $n$ that satisfies the inequality.

Conclusion

In conclusion, the solution to the inequality $3-|4-7|\ \textgreater \ 1$ is A. No solution. This is because the inequality $0\ \textgreater \ 1$ is never true, and therefore, there is no value of $n$ that satisfies the inequality.

Final Answer

The final answer is A. No solution.

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Q: What is the main concept behind solving the inequality $3-|4-7|\ \textgreater \ 1$?

A: The main concept behind solving the inequality $3-|4-7|\ \textgreater \ 1$ is to simplify the expression inside the absolute value and then evaluate the inequality.

Q: How do I simplify the expression inside the absolute value?

A: To simplify the expression inside the absolute value, you need to follow the order of operations (PEMDAS) and evaluate the expression inside the absolute value first.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that tells you which operations to perform first when you have multiple operations in an expression. The acronym PEMDAS stands for:

1. Parentheses: Evaluate expressions inside parentheses first.
2. Exponents: Evaluate any exponential expressions next.
3. Multiplication and Division: Evaluate any multiplication and division operations from left to right.
4. Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I evaluate the absolute value?

A: To evaluate the absolute value, you need to determine the distance of the expression inside the absolute value from zero. If the expression is positive, the absolute value is the same as the expression. If the expression is negative, the absolute value is the negative of the expression.

Q: What is the solution to the inequality $3-|4-7|\ \textgreater \ 1$?

A: The solution to the inequality $3-|4-7|\ \textgreater \ 1$ is A. No solution. This is because the inequality $0\ \textgreater \ 1$ is never true, and therefore, there is no value of $n$ that satisfies the inequality.

Q: Why is the solution A. No solution?

A: The solution is A. No solution because the inequality $0\ \textgreater \ 1$ is never true. This means that there is no value of $n$ that satisfies the inequality.

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 absolute value correctly
• Not simplifying the expression inside the absolute value
• Not checking the solution to the inequality

Q: How can I practice solving inequalities?

A: You can practice solving inequalities by working through examples and exercises in your textbook or online resources. You can also try solving inequalities on your own and checking your solutions with a calculator or online tool.

Q: What are some real-world applications of solving inequalities?

A: Solving inequalities has many real-world applications, including:

• Finance: Inequalities are used to calculate interest rates and investment returns.
• Science: Inequalities are used to model population growth and decay.
• Engineering: Inequalities are used to design and optimize systems.
• Economics: Inequalities are used to model economic systems and make predictions about future trends.

Q: Why is it important to understand how to solve inequalities?

A: Understanding how to solve inequalities is important because it allows you to model and analyze real-world problems. Inequalities are used in many fields, including finance, science, engineering, and economics, and being able to solve them is a valuable skill.