https://file.aiquickdraw.com/mathbot/file/1730238849242rv04j6o8.jpg

To solve the inequality:

\[
|x| + 3 > 9
\]

we'll isolate the absolute value expression and consider both possible cases for \( |x| \) being greater than a value.

### Step 1: Isolate \( |x| \)
Subtract 3 from both sides:

\[
|x| > 6
\]

### Step 2: Break into Two Cases
The inequality \( |x| > 6 \) means that \( x \) is either greater than 6 or less than -6. So we have:

1. \( x > 6 \)
2. \( x < -6 \)

### Solution
The solution to the inequality is:

\[
x > 6 \quad \text{or} \quad x < -6
\]

In interval notation, this can be written as:

\[
x \in (-\infty, -6) \cup (6, \infty)
\]

Would you like further details or have any questions?

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Here are some related questions that expand on this concept:

1. How do we solve inequalities that involve both absolute values and quadratic expressions?
2. What changes if we have \( |x| + 3 < 9 \) instead?
3. How do we solve compound inequalities involving absolute values, such as \( |x - 2| + 3 > 9 \)?
4. What is the graphical interpretation of absolute value inequalities?
5. How can absolute value inequalities be applied in real-world contexts?

**Tip:** Remember that absolute values produce two cases in inequalities, corresponding to both positive and negative scenarios.

Learn how to solve the absolute value inequality |x| + 3 > 9 with a step-by-step solution. This problem explores algebraic concepts involving absolute values and inequalities, ideal for students in Grades 9-10.

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