To solve the equation:

\[
-5 \cdot |3x + 7| + 4 = 14
\]

we need to isolate the absolute value expression, \(|3x + 7|\). Let's go step-by-step to solve for \(x\).

### Step 1: Isolate the absolute value

Subtract 4 from both sides:

\[
-5 \cdot |3x + 7| = 14 - 4 \implies -5 \cdot |3x + 7| = 10
\]

Divide both sides by \(-5\) to isolate the absolute value:

\[
|3x + 7| = -2
\]

### Step 2: Analyze the absolute value

Notice that the absolute value, \(|3x + 7|\), represents a non-negative quantity (always greater than or equal to 0). However, we have:

\[
|3x + 7| = -2
\]

Since an absolute value cannot be negative, this equation has **no solution**.

### Conclusion

The given equation has no valid solutions:

\[
\boxed{\text{No solution}}
\]

Would you like more details, or do you have any questions?

---

Here are 5 related questions to expand your understanding:

1. How do you solve equations involving absolute values when the result is positive?
2. Can an absolute value equation have exactly one solution? If so, under what conditions?
3. What does it mean graphically when an absolute value equation has no solution?
4. How do you solve the equation \( |ax + b| = c \) when \(c \geq 0\)?
5. What steps should you follow to solve equations with multiple absolute values?

**Tip:** Always check if the absolute value equation is valid by ensuring that the right side of the equation is non-negative.

Learn how to solve the absolute value equation -5 times the absolute value of 3x + 7 plus 4 equals 14 with detailed step-by-step solutions. This problem covers equations involving absolute values and linear equations. Suitable for students in Grades 9-12.

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