Okay, so I need to solve the inequality -4 < 2(x - 1) < 8 and express the solution in the form a < x < b, then find a + b. Hmm, let's start by recalling how to solve compound inequalities. I think the key is to isolate the variable x in the middle. Let me break it down step by step.

First, the inequality is -4 < 2(x - 1) < 8. Since it's a compound inequality, I can treat it as two separate inequalities connected by the variable term. Maybe I can split it into two parts: -4 < 2(x - 1) and 2(x - 1) < 8. Alternatively, I can treat the entire inequality as a single expression and manipulate it all at once. Let me try both approaches to see which is more straightforward.

Starting with the entire inequality: -4 < 2(x - 1) < 8. To isolate x, I should first divide all parts of the inequality by 2. Dividing by a positive number doesn't change the direction of the inequalities, so that's safe. Let's do that:

-4 ÷ 2 < (2(x - 1)) ÷ 2 < 8 ÷ 2

Simplifying each part:

-2 < x - 1 < 4

Now, to solve for x, I need to get rid of the -1 on the left side. I can add 1 to all parts of the inequality:

-2 + 1 < x - 1 + 1 < 4 + 1

Which simplifies to:

-1 < x < 5

So the solution is -1 < x < 5. Therefore, a is -1 and b is 5. The problem asks for a + b, so that's -1 + 5 = 4. Wait, is that right? Let me check my steps again to make sure I didn't make a mistake.

First, original inequality: -4 < 2(x - 1) < 8. Divided by 2: -2 < x - 1 < 4. Then added 1: -1 < x < 5. Yes, that seems correct. Alternatively, if I solve each part separately, let's see:

First inequality: -4 < 2(x - 1)

Divide by 2: -2 < x - 1

Add 1: -1 < x

Second inequality: 2(x - 1) < 8

Divide by 2: x - 1 < 4

Add 1: x < 5

So combining both, -1 < x < 5. Same result. So a is -1, b is 5. Sum is 4. That seems straightforward. Wait, but let me double-check by plugging in a value for x in the original inequality to see if it holds.

Let's take x = 0, which is between -1 and 5.

Original inequality: -4 < 2(0 - 1) < 8

Calculates to: -4 < 2(-1) < 8 → -4 < -2 < 8. Yes, that's true.

Another test: x = -2, which is less than -1. Let's see:

-4 < 2(-2 - 1) < 8 → -4 < 2(-3) < 8 → -4 < -6 < 8. Wait, -6 is not greater than -4. So that's false. So x = -2 is not in the solution, which is correct because the solution starts at -1.

Another test: x = 6, which is greater than 5.

2(6 - 1) = 2(5) = 10. So original inequality: -4 < 10 < 8? No, 10 is not less than 8. So x = 6 is not in the solution, which is correct. So the solution interval is indeed -1 to 5.

Therefore, a + b = -1 + 5 = 4. I think that's the answer. But let me just think if there's another way someone might approach this problem and possibly make a mistake.

Wait, sometimes when solving inequalities, especially when multiplying or dividing by variables, people might get confused about the direction of the inequality. But here, since we're dividing by 2, which is positive, the direction remains the same. So that's safe. So no issues there.

Alternatively, if someone tried to distribute the 2 first, but that would complicate things. Let's see:

Original inequality: -4 < 2(x - 1) < 8

If someone distributes the 2: -4 < 2x - 2 < 8

Then, add 2 to all parts: -4 + 2 < 2x < 8 + 2 → -2 < 2x < 10

Then divide by 2: -1 < x < 5. Same result. So either way, same answer. So regardless of the approach, the result is consistent. Therefore, the answer is 4.

I think that's solid. All steps check out, and the tests confirm the solution. So a + b is 4.
Therefore, the value of $a + b$ is $\boxed{4}$.
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