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Let's solve the given inequality step by step:

The inequality is:

\[
5 |x + 1| < 45
\]

### Step 1: Isolate the absolute value.
To simplify the inequality, divide both sides by 5:

\[
|x + 1| < 9
\]

### Step 2: Solve the absolute value inequality.
For the inequality \(|x + 1| < 9\), this means that the expression inside the absolute value, \(x + 1\), is between -9 and 9. Therefore, we can rewrite this as:

\[
-9 < x + 1 < 9
\]

### Step 3: Solve for \(x\).
Subtract 1 from all parts of the inequality:

\[
-9 - 1 < x < 9 - 1
\]

This simplifies to:

\[
-10 < x < 8
\]

### Step 4: Match the solution with the given choices.
The correct answer is:

\[
-10 < x < 8
\]

Thus, the correct option is the fourth one: **\(-10 < x < 8\)**.

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Would you like further clarification on any step?

Here are five related questions to explore:
1. How do you solve absolute value inequalities involving "greater than"?
2. What are the properties of absolute value functions in inequalities?
3. Can you explain how to graph the solution \(-10 < x < 8\) on a number line?
4. How would the solution change if the inequality had \(|x + 1| \leq 9\)?
5. How can absolute value inequalities be applied in real-world problems?

**Tip:** When solving absolute value inequalities, always remember to break them into two separate inequalities (one for the positive and one for the negative case).

What is the solution to the inequality? 5|x + 1| < 45

Learn how to solve the absolute value inequality 5|x + 1| < 45 with detailed steps. This problem involves isolating the absolute value, breaking it into a double inequality, and solving for x. Suitable for students in Grades 8-10.

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