Solution: The equation simplifies to $|b - 3| = 5$, so $b - 3 = 5$ or $b - 3 = -5$. Solving gives $b = 8$ or $b = -2$. Sum: $8 + (-2) = 6$. $\boxed{6}$

["Understanding the Equation: Solving $|b - 3| = 5$ and Finding the Sum of Solutions", "Facing a simple absolute value equation like $|b - 3| = 5$ can feel straightforward, but solving it correctly is essential for mastering algebra. This article breaks down the step-by-step solution process and reveals how to find the sum of the solutions efficiently.", "### What is the Absolute Value Equation?", "The equation $|b - 3| = 5$ means that the expression inside the absolute value, $b - 3$, is exactly 5 units away from zero on the number line. By definition, the absolute value equation $|x| = a$ has two possible solutions when $a > 0$:\n- $x = a$\n- $x = -a$", "Applying this to our problem:\n$$\n|b - 3| = 5 \Rightarrow b - 3 = 5 \quad \ ext{or} \quad b - 3 = -5\n$$", "### Step 1: Solve Each Case", "Case 1:\n$$\nb - 3 = 5 \Rightarrow b = 5 + 3 = 8\n$$", "Case 2:\n$$\nb - 3 = -5 \Rightarrow b = -5 + 3 = -2\n$$", "### Step 2: Find the Sum of the Solutions", "The two valid solutions are $b = 8$ and $b = -2$. To find their sum:\n$$\n8 + (-2) = 6\n$$", "Thus, the sum of all solutions is $\boxed{6}$.", "### Why This Matters", "Solving absolute value equations like $|b - 3| = 5$ is foundational for many real-world applications—from computing distances on a number line to solving mathematical modeling problems. Understanding how to break down and simplify such equations ensures accurate results and strengthens algebraic reasoning.", "In summary, the equation simplifies neatly to two linear solutions, and their sum is $\boxed{6}$. Whether you're studying for a test, working through homework, or exploring math concepts, mastering absolute value equations is a valuable skill!"]