Let's solve the compound inequality step by step:

First inequality:

$\frac{b}{3} - 5 \geq 2$

1. Add 5 to both sides: $\frac{b}{3} \geq 7$
2. Multiply both sides by 3: $b \geq 21$

Second inequality:

$\frac{5 - b}{4} > \frac{1}{2}$

1. Multiply both sides by 4 to eliminate the denominator: $5 - b > 2$
2. Subtract 5 from both sides: $-b > -3$
3. Multiply both sides by -1 (remember to reverse the inequality sign): $b < 3$

Compound inequality:

From the two inequalities, we have: $b \geq 21 \quad \text{or} \quad b < 3$

Thus, the solution is: $b < 3 \quad \text{or} \quad b \geq 21$

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

Here are 5 related questions:

1. How do you solve inequalities with fractions and variables?
2. What is the rule for reversing the inequality sign when multiplying by a negative number?
3. How can you graph a compound inequality on a number line?
4. What is the difference between "and" and "or" in compound inequalities?
5. Can compound inequalities ever have overlapping solutions?

Tip: Always pay attention to the direction of the inequality when multiplying or dividing by negative numbers!