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To solve the inequality \(|3 - 2x| > 11\), we need to break it down into two separate cases, since the absolute value inequality can be expressed as:

\[
3 - 2x > 11 \quad \text{or} \quad 3 - 2x < -11.
\]

Let's solve each case individually:

### Case 1: \(3 - 2x > 11\)

1. Subtract 3 from both sides:
   \[
   -2x > 8
   \]
2. Divide by -2 (remember to flip the inequality sign when dividing by a negative number):
   \[
   x < -4
   \]

### Case 2: \(3 - 2x < -11\)

1. Subtract 3 from both sides:
   \[
   -2x < -14
   \]
2. Divide by -2 (again, flip the inequality sign):
   \[
   x > 7
   \]

### Solution Set
The combined solution is \(x < -4\) or \(x > 7\). In interval notation, this is:

\[
(-\infty, -4) \cup (7, \infty)
\]

Thus, the correct answer is option A, with the solution set being \((-\infty, -4) \cup (7, \infty)\).

Would you like more details or have any questions?

Here are some related questions to explore:

1. How do you solve absolute value inequalities in general?
2. How would the solution change if the inequality were \(|3 - 2x| \leq 11\)?
3. What happens if the inequality were \(|3 - 2x| < -11\)? Is there any real solution?
4. How would you graph the solution set on a number line?
5. Can you solve \(|2x - 3| > 5\) using a similar method?

**Tip:** When solving absolute value inequalities, remember to split them into two cases based on the definition of absolute value, and always flip the inequality when multiplying or dividing by a negative number.

Solve the inequality |3 - 2x| > 11 and graph the solution set.

Learn how to solve the absolute value inequality |3 - 2x| > 11 step-by-step. This solution covers key algebraic concepts, including solving inequalities and the properties of absolute value. Suitable for students in Grades 9-12.

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