Settlement Of Inequality 2 ≤ | X-4 | ≤ 3

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Introduction

In mathematics, inequalities are used to describe the relationship between two or more values. They are an essential part of mathematical expressions and are used to solve various problems in algebra, geometry, and other branches of mathematics. In this article, we will focus on the settlement of the inequality 2 ≤ | X-4 | ≤ 3, which involves absolute values and inequalities.

Understanding Absolute Values

Before we dive into the settlement of the inequality, it's essential to understand what absolute values are. Absolute values are a mathematical concept that represents the distance of a number from zero on the number line. In other words, it's the magnitude of a number without considering its sign. The absolute value of a number x is denoted by |x| and is always non-negative.

The Inequality 2 ≤ | X-4 | ≤ 3

The given inequality is 2 ≤ | X-4 | ≤ 3. This inequality involves absolute values and can be solved using the properties of absolute values. To solve this inequality, we need to consider two cases: when X-4 is positive and when X-4 is negative.

Case 1: X-4 ≥ 0

When X-4 ≥ 0, we can rewrite the inequality as 2 ≤ X-4 ≤ 3. To solve this inequality, we need to isolate the variable X. We can do this by adding 4 to all parts of the inequality.

2 + 4 ≤ X - 4 + 4 ≤ 3 + 4
6 ≤ X ≤ 7

Case 2: X-4 < 0

When X-4 < 0, we can rewrite the inequality as -3 ≤ X-4 ≤ -2. To solve this inequality, we need to isolate the variable X. We can do this by adding 4 to all parts of the inequality.

-3 + 4 ≤ X - 4 + 4 ≤ -2 + 4
1 ≤ X ≤ 2

Combining the Cases

Now that we have solved the inequality for both cases, we can combine the results to find the final solution. The solution to the inequality 2 ≤ | X-4 | ≤ 3 is the union of the two cases.

6 ≤ X ≤ 7 or 1 ≤ X ≤ 2

Conclusion

In this article, we have solved the inequality 2 ≤ | X-4 | ≤ 3 using the properties of absolute values. We have considered two cases: when X-4 is positive and when X-4 is negative. By isolating the variable X in each case, we have found the solution to the inequality. The final solution is the union of the two cases, which is 6 ≤ X ≤ 7 or 1 ≤ X ≤ 2.

Frequently Asked Questions

  • What is the meaning of the absolute value of a number? The absolute value of a number is its magnitude without considering its sign.
  • How do you solve an inequality involving absolute values? To solve an inequality involving absolute values, you need to consider two cases: when the expression inside the absolute value is positive and when it is negative.
  • What is the solution to the inequality 2 ≤ | X-4 | ≤ 3? The solution to the inequality 2 ≤ | X-4 | ≤ 3 is 6 ≤ X ≤ 7 or 1 ≤ X ≤ 2.

Final Thoughts

In conclusion, solving inequalities involving absolute values requires careful consideration of the properties of absolute values. By understanding the concept of absolute values and using the properties of inequalities, we can solve complex inequalities like 2 ≤ | X-4 | ≤ 3. This article has provided a step-by-step guide to solving this inequality, and we hope that it has been helpful in understanding the concept of absolute values and inequalities.

Introduction

In our previous article, we solved the inequality 2 ≤ | X-4 | ≤ 3 using the properties of absolute values. We considered two cases: when X-4 is positive and when X-4 is negative. By isolating the variable X in each case, we found the solution to the inequality. In this article, we will provide a Q&A section to help clarify any doubts or questions that readers may have.

Q&A

Q: What is the meaning of the absolute value of a number?

A: The absolute value of a number is its magnitude without considering its sign. In other words, it's the distance of a number from zero on the number line.

Q: How do you solve an inequality involving absolute values?

A: To solve an inequality involving absolute values, you need to consider two cases: when the expression inside the absolute value is positive and when it is negative. You then isolate the variable in each case to find the solution to the inequality.

Q: What is the solution to the inequality 2 ≤ | X-4 | ≤ 3?

A: The solution to the inequality 2 ≤ | X-4 | ≤ 3 is 6 ≤ X ≤ 7 or 1 ≤ X ≤ 2.

Q: Can you explain the concept of absolute values in more detail?

A: Absolutely! The absolute value of a number x is denoted by |x| and is always non-negative. It represents the distance of a number from zero on the number line. For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5.

Q: How do you handle negative numbers when working with absolute values?

A: When working with absolute values, you can ignore the negative sign and focus on the magnitude of the number. For example, |-5| = 5, and |5| = 5.

Q: Can you provide more examples of solving inequalities involving absolute values?

A: Yes, certainly! Here are a few more examples:

  • |X-2| ≤ 3: This inequality can be solved by considering two cases: when X-2 is positive and when X-2 is negative. The solution to this inequality is -1 ≤ X ≤ 5.
  • |X+1| ≥ 2: This inequality can be solved by considering two cases: when X+1 is positive and when X+1 is negative. The solution to this inequality is X ≤ -3 or X ≥ 1.

Q: What are some common mistakes to avoid when solving inequalities involving absolute values?

A: Some common mistakes to avoid when solving inequalities involving absolute values include:

  • Failing to consider both cases (when the expression inside the absolute value is positive and when it is negative)
  • Not isolating the variable in each case
  • Not checking the solution to ensure it satisfies the original inequality

Conclusion

In this Q&A article, we have provided answers to some common questions about solving inequalities involving absolute values. We have also provided additional examples and tips to help readers better understand the concept of absolute values and inequalities. By following the steps outlined in this article, readers should be able to solve complex inequalities like 2 ≤ | X-4 | ≤ 3 with confidence.

Final Thoughts

Solving inequalities involving absolute values requires careful consideration of the properties of absolute values. By understanding the concept of absolute values and using the properties of inequalities, we can solve complex inequalities like 2 ≤ | X-4 | ≤ 3. This article has provided a step-by-step guide to solving this inequality, and we hope that it has been helpful in understanding the concept of absolute values and inequalities.