We are tasked with proving the inequality:

\[
3 \cdot |x - 4| - 2 \cdot |x + 7| \geq -22
\]

### Steps for the solution:

1. **Rewrite the inequality**:
   Start by adding \(22\) to both sides of the inequality:
   
   \[
   3 \cdot |x - 4| - 2 \cdot |x + 7| + 22 \geq 0
   \]

2. **Split into cases**:
   We need to split this inequality into cases based on the critical points where the absolute values change, i.e., at \(x = 4\) and \(x = -7\).

### Case 1: \(x \geq 4\)
   In this region, both \(x - 4 \geq 0\) and \(x + 7 \geq 0\), so the absolute values drop, and we can rewrite the inequality as:

   \[
   3(x - 4) - 2(x + 7) \geq -22
   \]

   Simplifying this:

   \[
   3x - 12 - 2x - 14 \geq -22
   \]

   Combine like terms:

   \[
   x - 26 \geq -22
   \]

   Add 26 to both sides:

   \[
   x \geq 4
   \]

   This is true for all \(x \geq 4\), which is the range we are already considering. Hence, the inequality holds for \(x \geq 4\).

### Case 2: \(-7 \leq x < 4\)
   In this region, \(x - 4 < 0\) and \(x + 7 \geq 0\), so we modify the absolute values accordingly:

   \[
   3(4 - x) - 2(x + 7) \geq -22
   \]

   Simplify:

   \[
   12 - 3x - 2x - 14 \geq -22
   \]

   Combine like terms:

   \[
   -5x - 2 \geq -22
   \]

   Add 2 to both sides:

   \[
   -5x \geq -20
   \]

   Divide by -5 (remember to reverse the inequality):

   \[
   x \leq 4
   \]

   This is valid since we are in the range \(-7 \leq x < 4\).

### Case 3: \(x < -7\)
   In this region, both \(x - 4 < 0\) and \(x + 7 < 0\), so we modify the absolute values:

   \[
   3(4 - x) - 2(-x - 7) \geq -22
   \]

   Simplify:

   \[
   12 - 3x + 2x + 14 \geq -22
   \]

   Combine like terms:

   \[
   -x + 26 \geq -22
   \]

   Subtract 26 from both sides:

   \[
   -x \geq -48
   \]

   Multiply by -1 (reverse the inequality):

   \[
   x \leq 48
   \]

   This is always true in the region \(x < -7\).

### Conclusion:

The inequality holds for all \(x\). Therefore, the original inequality is valid.

Do you want more details or have any questions?

Here are 5 related questions:
1. What are the properties of absolute value functions in inequalities?
2. How do we solve inequalities involving multiple absolute values?
3. Can absolute values lead to systems of inequalities?
4. What is the geometric interpretation of absolute values in inequalities?
5. How does dividing by negative numbers affect inequalities?

**Tip:** When working with absolute values, always consider different cases based on the points where expressions inside the absolute values change signs.

Learn how to solve the inequality 3|x - 4| - 2|x + 7| >= -22 by splitting into cases based on absolute values. This step-by-step solution covers algebraic concepts and absolute value properties. Suitable for students in Grades 9-12.

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