4|z+2|<32
Step 1 of 2 :  Solve the inequality and express your answer in interval notation. Use decimal form for numerical values.

To solve the inequality \( 4|z + 2| < 32 \), follow these steps:

### Step 1: Isolate the absolute value expression.
Divide both sides of the inequality by 4:

\[
|z + 2| < \frac{32}{4}
\]

\[
|z + 2| < 8
\]

### Step 2: Write the compound inequality.
The inequality \( |z + 2| < 8 \) means that the expression inside the absolute value lies between \(-8\) and \(8\):

\[
-8 < z + 2 < 8
\]

### Step 3: Solve the compound inequality.
Subtract 2 from all parts of the inequality:

\[
-8 - 2 < z < 8 - 2
\]

\[
-10 < z < 6
\]

### Step 4: Express the solution in interval notation.
The solution to the inequality is all values of \( z \) between \(-10\) and \(6\). In interval notation, this is:

\[
(-10, 6)
\]

So, the solution in interval notation is \( (-10, 6) \).

Would you like to go over the details or have any questions? Here are five related questions to further your understanding:

1. How would the solution change if the inequality were \( 4|z + 2| \leq 32 \)?
2. What would the interval notation be if the inequality were \( |z + 2| > 8 \)?
3. How do you solve an absolute value inequality of the form \( |ax + b| < c \)?
4. What happens if the inequality has a negative number on the right side, such as \( 4|z + 2| < -32 \)?
5. Can you solve the inequality \( |z - 3| + 5 < 12 \)?

**Tip:** When solving absolute value inequalities, always start by isolating the absolute value expression before creating a compound inequality.

Learn how to solve the inequality 4|z + 2| < 32 step-by-step with detailed explanations and interval notation. This problem covers absolute value inequalities and is suitable for high school students.

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