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The problem asks for all \( x \)-values within a distance of 10 from the number 6, and it requests the answer in interval notation.

### Step-by-Step Solution:

1. The condition "within a distance of 10 from the number 6" can be written as an absolute value inequality:
   \[
   |x - 6| \leq 10
   \]
   
2. To solve this absolute value inequality, we break it into two separate inequalities:
   \[
   -10 \leq x - 6 \leq 10
   \]

3. Now, solve for \( x \) by adding 6 to all parts of the inequality:
   \[
   -10 + 6 \leq x \leq 10 + 6
   \]
   \[
   -4 \leq x \leq 16
   \]

4. The solution in interval notation is:
   \[
   [-4, 16]
   \]

So, the values of \( x \) that are within a distance of 10 from the number 6 are in the interval \( [-4, 16] \).

Would you like further details or have any other questions?

Here are 5 related questions:
1. How do you solve absolute value inequalities?
2. What is the graphical interpretation of \( |x - a| \leq b \)?
3. How do you express inequalities involving absolute values as compound inequalities?
4. What if the distance in the problem was less than 10 instead of equal to 10?
5. Can absolute value inequalities have more than one solution?

**Tip:** Always remember, absolute value represents the distance from zero, and inequalities involving absolute values can often be split into two linear inequalities.

Describe all x-values within a distance of 10 from the number 6. Enter your answer in interval notation.

Learn how to solve an absolute value inequality describing x-values within a distance of 10 from the number 6. This step-by-step solution explains key algebraic concepts, absolute value inequalities, and compound inequalities. Suitable for Grades 7-10.

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