The given inequality is:

$|10 + 5b| < 30$

To solve it, we'll break this absolute value inequality into two cases:

1. $10 + 5b < 30$
2. $-(10 + 5b) < 30$ (which is equivalent to $10 + 5b > -30$)

Case 1:

$10 + 5b < 30$ Subtract 10 from both sides: $5b < 20$ Now divide by 5: $b < 4$

Case 2:

$-(10 + 5b) < 30$ This simplifies to: $10 + 5b > -30$ Subtract 10 from both sides: $5b > -40$ Now divide by 5: $b > -8$

Combined Solution:

From both cases, the solution is: $-8 < b < 4$

This means $b$ lies between $-8$ and $4$.

Would you like more details or have any questions?

Related Questions:

1. How do we solve inequalities with absolute values in general?
2. What are the properties of absolute value functions?
3. How do you graph solutions of compound inequalities?
4. What is the difference between strict inequalities and non-strict inequalities?
5. How does the solution change if the inequality were $|10 + 5b| \leq 30$?

Tip:

Always remember that when dealing with absolute values, you are considering both the positive and negative possibilities of the expression inside.