The number line shows values from -8 to 6, inclusive, indicated by two closed dots at these positions. We need to determine which of the given absolute value expressions correctly describes this range.

Step-by-step solution:

• The interval of values is from $-8$ to $6$. In absolute value notation, an interval can be expressed as a center point plus or minus a certain distance.

Let’s analyze the choices:

1. Option A: $|x - 2| \leq 6$

   The expression $|x - 2| \leq 6$ represents the interval of all numbers that are within 6 units of 2. This means the interval is: $2 - 6 \leq x \leq 2 + 6 \quad \Rightarrow \quad -4 \leq x \leq 8$ This does not match the number line shown, since the interval is from -8 to 6, not -4 to 8.

2. Option B: $|x - 2| \geq -6$

   This inequality does not make sense, because absolute values are always non-negative, and an absolute value being greater than or equal to a negative number is always true. Therefore, this does not represent a specific interval.

3. Option C: $|x - 6| \geq 2$

   The expression $|x - 6| \geq 2$ represents values that are at least 2 units away from 6. This would create two intervals: $x \leq 4 \quad \text{or} \quad x \geq 8$ This does not match the given interval on the number line.

4. Option D: $|x - 6| \geq -2$

   Similar to Option B, this inequality doesn't make sense because an absolute value is always greater than or equal to 0, so this statement is always true.

5. Option E: $|x + 6| \geq 2$

   The expression $|x + 6| \geq 2$ represents values that are at least 2 units away from -6. This creates two intervals: $x \leq -8 \quad \text{or} \quad x \geq -4$ This matches part of the number line, but it does not cover the entire interval from -8 to 6.

Correct Answer:

None of the options match the interval on the number line perfectly.

Would you like further clarification or additional details? Here are some related questions you might find useful:

1. How do you graph inequalities on a number line?
2. How do you convert absolute value inequalities into interval notation?
3. What is the general rule for solving absolute value inequalities?
4. Can absolute values ever be less than negative numbers?
5. How can absolute value inequalities represent disjoint intervals?

Tip: When working with absolute value inequalities, it's helpful to rewrite the inequality as two separate linear inequalities to fully understand the range they represent.