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Let's solve the given inequality:

### Given:
\[
4(3 - b) - 3(4 - b) > -15
\]

### Step 1: Expand the expressions.
Distribute the constants inside the parentheses.

\[
4 \cdot 3 - 4 \cdot b - 3 \cdot 4 + 3 \cdot b > -15
\]

\[
12 - 4b - 12 + 3b > -15
\]

### Step 2: Simplify the equation.
Combine like terms.

\[
-4b + 3b > -15
\]

\[
-b > -15
\]

### Step 3: Solve for \(b\).
Divide both sides by \(-1\). **Remember to reverse the inequality sign** when dividing by a negative number.

\[
b < 15
\]

### Final Answer:
The correct inequality is:

\[
b < 15
\]

Thus, the correct answer is **the first option**: \(b < 15\).

---

Would you like further details or have any other questions?

### 5 Related Questions:
1. What happens if you forget to reverse the inequality sign when dividing by a negative number?
2. How would the solution change if the inequality were \( \geq -15 \) instead?
3. How do you know when to distribute vs. simplify directly in inequalities?
4. How do inequalities change when dealing with absolute values?
5. What are some tips for solving compound inequalities?

### Tip:
Always double-check for sign changes when multiplying or dividing by negative numbers!

Learn how to solve the linear inequality 4(3 - b) - 3(4 - b) > -15. This step-by-step solution covers key algebraic concepts, such as the distributive property and solving inequalities with negative coefficients. Suitable for students in Grades 8-10.

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