Solve for z.
|z|+1<3
Write a compound inequality like 1 < x < 3 or like x < 1 or x > 3. Use integers, proper fractions, or improper fractions in simplest form.

We start with the inequality:

\[
|z| + 1 < 3
\]

### Step 1: Isolate the absolute value term
Subtract 1 from both sides:

\[
|z| < 2
\]

### Step 2: Remove the absolute value
The inequality \(|z| < 2\) means that \(z\) must be between \(-2\) and \(2\). So, we can write the compound inequality as:

\[
-2 < z < 2
\]

This is the solution in compound inequality form.

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Would you like any additional details or have questions about this solution?

Here are 5 related questions:
1. How would the inequality change if the original problem were \(|z| + 1 \leq 3\)?
2. What if the inequality was \(|z| + 1 > 3\)? How would the solution look?
3. How do we solve absolute value equations instead of inequalities?
4. Can the inequality have solutions if it was \(|z| + 1 = 3\)?
5. How does solving absolute value inequalities differ from solving absolute value equations?

**Tip:** When solving absolute value inequalities, remember that \(|x| < a\) leads to a compound inequality \(-a < x < a\), while \(|x| > a\) leads to two separate inequalities, \(x > a\) or \(x < -a\).

|z| + 1 < 3. Write a compound inequality like 1 < x < 3 or like x < 1 or x > 3. Use integers, proper fractions, or improper fractions in simplest form.

Learn how to solve the absolute value inequality |z| + 1 < 3 by isolating the absolute value and removing it to form a compound inequality. This step-by-step solution is suitable for students in Grades 6-8 and covers essential algebraic concepts such as absolute value and inequalities.

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