The problem involves solving and graphing the compound inequality $-8 < x - 6 < 10$. Let's break it down step by step.

Step 1: Solve the compound inequality

The compound inequality can be split into two separate inequalities:

1. $-8 < x - 6$
2. $x - 6 < 10$

Inequality 1: $-8 < x - 6$

• Add 6 to both sides to isolate $x$: $-8 + 6 < x$ $-2 < x$ This simplifies to: $x > -2$

Inequality 2: $x - 6 < 10$

• Add 6 to both sides to isolate $x$: $x - 6 + 6 < 10 + 6$ $x < 16$

Step 2: Combine the inequalities

Now that we have $x > -2$ and $x < 16$, we combine them to get: $-2 < x < 16$

Step 3: Write the solution in interval notation

The solution set in interval notation is: $(-2, 16)$

Step 4: Graph the solution

On a number line, the solution would be represented as an open interval from $-2$ to $16$, meaning $x$ can take any value between $-2$ and $16$ but not including $-2$ and $16$.

Answer

So the correct solution set is $(-2, 16)$.

If you need any more details or have any questions, feel free to ask!

Here are 5 related questions that might help expand your understanding:

1. What would be the solution set if the inequality was $-8 \leq x - 6 \leq 10$?
2. How would the solution change if the inequality was $-8 < x - 6 \leq 10$?
3. Can you graph the solution set on a number line?
4. What does it mean when the solution set is empty (i.e., $\emptyset$)?
5. How would the interval notation change if the inequalities were "greater than or equal to" instead of "greater than"?

Tip: When working with compound inequalities, solve each part separately and then combine the results to find the solution set.