given inequality |3/2 z-1|_< 2 is becoming in pair of

We are given the inequality:

\[
\left| \frac{3}{2} z - 1 \right| < 2
\]

To solve this, we will break the absolute value inequality into two inequalities and solve for \( z \).

### Step 1: Remove the absolute value
Recall that for any expression \( |A| < B \), it implies:

\[
-B < A < B
\]

So, for the given inequality:

\[
-2 < \frac{3}{2} z - 1 < 2
\]

### Step 2: Solve for \( z \)
Now, solve each part of this compound inequality.

#### Solve the left side:
\[
-2 < \frac{3}{2} z - 1
\]
Add 1 to both sides:
\[
-1 < \frac{3}{2} z
\]
Multiply both sides by \( \frac{2}{3} \):
\[
-\frac{2}{3} < z
\]

#### Solve the right side:
\[
\frac{3}{2} z - 1 < 2
\]
Add 1 to both sides:
\[
\frac{3}{2} z < 3
\]
Multiply both sides by \( \frac{2}{3} \):
\[
z < 2
\]

### Step 3: Combine the inequalities
We now combine the two results:

\[
-\frac{2}{3} < z < 2
\]

Thus, the solution to the inequality is:

\[
z \in \left( -\frac{2}{3}, 2 \right)
\]

Would you like more details or have any questions?

### Related Questions:
1. How do you solve absolute value inequalities of the form \( |A| > B \)?
2. Can you explain how to graph the solution of this inequality on a number line?
3. What changes in the solution if the inequality is \( \left| \frac{3}{2} z - 1 \right| > 2 \)?
4. How would you handle an inequality with multiple absolute values?
5. What are the general steps for solving compound inequalities?

**Tip:** Always remember that solving absolute value inequalities typically involves breaking them into two separate inequalities.

given inequality |3/2 z-1| < 2 is becoming in pair of

Solve the absolute value inequality |3/2 z - 1| < 2 with a detailed, step-by-step approach. Learn how to break down and solve compound inequalities, covering key algebraic concepts and absolute value inequalities. Suitable for Grades 9-11 students.

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