Let's solve the inequality:

\[
|-1.5x - 2| + 1 < 6
\]

### Step 1: Isolate the absolute value expression
First, subtract 1 from both sides to isolate the absolute value:

\[
|-1.5x - 2| < 6 - 1
\]
\[
|-1.5x - 2| < 5
\]

### Step 2: Break down the absolute value inequality
An absolute value inequality of the form \(|A| < B\) can be rewritten as two inequalities:

\[
-5 < -1.5x - 2 < 5
\]

### Step 3: Solve the compound inequality
Now, solve each part of the compound inequality.

#### Part 1: Solve \( -5 < -1.5x - 2 \)
Add 2 to both sides:

\[
-5 + 2 < -1.5x
\]
\[
-3 < -1.5x
\]

Divide both sides by -1.5, and remember to flip the inequality sign because you're dividing by a negative number:

\[
\frac{-3}{-1.5} > x
\]
\[
2 > x
\]
Or:
\[
x < 2
\]

#### Part 2: Solve \( -1.5x - 2 < 5 \)
Add 2 to both sides:

\[
-1.5x < 5 + 2
\]
\[
-1.5x < 7
\]

Now divide by -1.5, flipping the inequality sign:

\[
x > \frac{7}{-1.5}
\]
\[
x > -\frac{14}{3} \quad \text{(which is approximately } x > -4.67\text{)}
\]

### Step 4: Combine the results
From both parts, we get:

\[
-4.67 < x < 2
\]

So, the solution to the inequality is:

\[
x \in \left( -\frac{14}{3}, 2 \right)
\]

Let me know if you want further clarification, or if you have any questions!

---

Here are 5 related questions for practice:

1. Solve: \(|2x + 3| - 4 \leq 7\)
2. Solve: \(|x - 5| > 3\)
3. Solve the quadratic inequality: \(x^2 - 4x - 5 < 0\)
4. Solve: \(3|x + 1| \geq 9\)
5. Solve the system of inequalities: \(2x - 3 > 1\) and \(4x + 1 \leq 9\)

**Tip**: When dealing with absolute value inequalities, always remember to consider both the positive and negative cases.

Learn how to solve the absolute value inequality |-1.5x - 2| + 1 < 6 using step-by-step algebraic techniques. This solution covers breaking down absolute value inequalities and solving compound linear inequalities, suitable for Grades 9-11.

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