To solve the inequality:

\[
-3 \left| 2 - 2x \right| - 2 < -32
\]

### Step 1: Isolate the absolute value term
First, move the constant \(-2\) to the other side:

\[
-3 \left| 2 - 2x \right| < -32 + 2
\]

Simplifying the right-hand side:

\[
-3 \left| 2 - 2x \right| < -30
\]

### Step 2: Divide by \(-3\)
When dividing by a negative number, we reverse the inequality sign:

\[
\left| 2 - 2x \right| > 10
\]

### Step 3: Break the absolute value into two cases
For \(\left| 2 - 2x \right| > 10\), this results in two inequalities:

1. \( 2 - 2x > 10 \)
2. \( 2 - 2x < -10 \)

### Step 4: Solve each inequality

#### Case 1: \( 2 - 2x > 10 \)
\[
2 - 2x > 10
\]
Subtract 2 from both sides:
\[
-2x > 8
\]
Divide by \(-2\) (and reverse the inequality sign):
\[
x < -4
\]

#### Case 2: \( 2 - 2x < -10 \)
\[
2 - 2x < -10
\]
Subtract 2 from both sides:
\[
-2x < -12
\]
Divide by \(-2\) (and reverse the inequality sign):
\[
x > 6
\]

### Step 5: Combine the solutions
The solution is:

\[
x < -4 \quad \text{or} \quad x > 6
\]

This means the inequality holds for \(x\) in the intervals \( (-\infty, -4) \cup (6, \infty) \).

---

Would you like more details or explanations? Here are some related questions to explore:

1. How do you solve inequalities involving absolute values?
2. Can inequalities have multiple intervals in their solutions?
3. Why do we reverse the inequality when dividing by a negative number?
4. How does removing absolute values affect the structure of an inequality?
5. What are the graphical representations of inequalities?

**Tip:** Always remember to reverse the inequality sign when multiplying or dividing by a negative number.

Learn how to solve the absolute value inequality −3|2−2x|−2<−32 with a detailed step-by-step guide. This algebraic problem involves manipulating absolute values and inequalities, using key concepts such as inequality reversal when dividing by negative numbers. Suitable for students in Grades 9-11.

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