What Is The Solution To $|x-6| \geq 5$?A. $1 \leq X \leq 11$ B. $-11 \leq X \leq 1$ C. $x \geq 11$ Or $x \leq 1$ D. $x \geq 1$ Or $x \leq -11$

Introduction

In mathematics, absolute value equations are a type of inequality that involves the absolute value of a variable or expression. These equations are used to represent the distance of a value from a certain point on the number line. In this article, we will explore the solution to the absolute value inequality $|x-6| \geq 5$. We will break down the steps to solve this inequality and provide the final solution.

Understanding Absolute Value Inequalities

Absolute value inequalities involve the absolute value of a variable or expression. The absolute value of a number is its distance from zero on the number line. For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5. When solving absolute value inequalities, we need to consider both the positive and negative cases.

Solving the Inequality $|x-6| \geq 5$

To solve the inequality $|x-6| \geq 5$, we need to consider two cases:

Case 1: $x-6 \geq 5$

In this case, we can add 6 to both sides of the inequality to get $x \geq 11$.

Case 2: $x-6 < -5$

In this case, we can add 6 to both sides of the inequality to get $x < -1$.

Combining the Cases

Since the absolute value inequality $|x-6| \geq 5$ involves both the positive and negative cases, we need to combine the two cases to get the final solution.

The solution to the inequality $|x-6| \geq 5$ is $x \geq 11$ or $x \leq -1$.

Checking the Solution

To check the solution, we can plug in values of $x$ that satisfy the inequality and verify that the absolute value of $x-6$ is indeed greater than or equal to 5.

For example, if we let $x=11$, we get $|11-6| = 5$, which satisfies the inequality. Similarly, if we let $x=-1$, we get $|-1-6| = 7$, which also satisfies the inequality.

Conclusion

In conclusion, the solution to the absolute value inequality $|x-6| \geq 5$ is $x \geq 11$ or $x \leq -1$. This solution represents the set of all values of $x$ that satisfy the inequality.

Final Answer

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

Introduction

In our previous article, we explored the solution to the absolute value inequality $|x-6| \geq 5$. In this article, we will answer some frequently asked questions (FAQs) about absolute value inequalities. These questions cover various topics, including the definition of absolute value, solving absolute value inequalities, and checking the solution.

Q&A

Q1: What is the definition of absolute value?

A1: The absolute value of a number is its distance from zero on the number line. For example, the absolute value of 5 is 5, and the absolute value of -5 is also 5.

Q2: How do I solve an absolute value inequality?

A2: To solve an absolute value inequality, you need to consider both the positive and negative cases. You can use the following steps:

• Write the inequality as two separate inequalities, one for the positive case and one for the negative case.
• Solve each inequality separately.
• Combine the solutions to get the final answer.

Q3: What is the difference between $|x-6| \geq 5$ and $|x-6| \leq 5$?

A3: The inequality $|x-6| \geq 5$ represents all values of $x$ that are greater than or equal to 5 units away from 6 on the number line. The inequality $|x-6| \leq 5$ represents all values of $x$ that are less than or equal to 5 units away from 6 on the number line.

Q4: How do I check the solution to an absolute value inequality?

A4: To check the solution, you can plug in values of $x$ that satisfy the inequality and verify that the absolute value of $x-6$ is indeed greater than or equal to 5 (or less than or equal to 5, depending on the inequality).

Q5: What is the solution to the inequality $|x+3| \geq 2$?

A5: To solve the inequality $|x+3| \geq 2$, we need to consider two cases:

• Case 1: $x+3 \geq 2$
• Case 2: $x+3 < -2$

Solving each case, we get:

• Case 1: $x \geq -1$
• Case 2: $x < -5$

Combining the cases, we get the final solution: $x \geq -1$ or $x < -5$.

Q6: What is the solution to the inequality $|x-2| \leq 3$?

A6: To solve the inequality $|x-2| \leq 3$, we need to consider two cases:

• Case 1: $x-2 \leq 3$
• Case 2: $x-2 > -3$

Solving each case, we get:

• Case 1: $x \leq 5$
• Case 2: $x > -1$

Combining the cases, we get the final solution: $-1 < x \leq 5$.

Conclusion

In conclusion, absolute value inequalities are an important topic in mathematics. By understanding the definition of absolute value and how to solve absolute value inequalities, you can solve a wide range of problems. We hope that this FAQ article has been helpful in answering your questions about absolute value inequalities.

Final Answer

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