The solution set in interval notation for |P-2| < 6 is

We are asked to solve the inequality \( |P - 2| < 6 \) and express the solution set in interval notation.

### Step 1: Understand the Absolute Value Inequality

The inequality \( |P - 2| < 6 \) means that the expression inside the absolute value, \( P - 2 \), is within 6 units of 0. This can be rewritten as a double inequality:

\[
-6 < P - 2 < 6
\]

### Step 2: Solve the Double Inequality

Now, solve for \( P \) by adding 2 to all three parts of the inequality:

\[
-6 + 2 < P < 6 + 2
\]
\[
-4 < P < 8
\]

### Step 3: Express the Solution in Interval Notation

The solution to the inequality is all values of \( P \) between -4 and 8, but not including -4 and 8, since the inequality is strict ("<").

Thus, the solution in interval notation is:

\[
(-4, 8)
\]

Let me know if you need more details or have any questions.

Here are 5 related questions for further exploration:
1. How do you solve inequalities involving absolute value and different constants?
2. What changes if the inequality is \( |P - 2| \leq 6 \)?
3. How would the solution change if the inequality was \( |P - 2| > 6 \)?
4. How do you graph the solution \( (-4, 8) \) on a number line?
5. What happens if the inequality involves both absolute value and a quadratic expression?

**Tip:** When solving absolute value inequalities, always break them into two cases (positive and negative) and solve them as compound inequalities.

Learn how to solve the absolute value inequality |P - 2| < 6 with a detailed step-by-step solution. This problem covers key algebraic concepts, including absolute value inequalities and the related theorem. Suitable for students in Grades 8-10.

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