We are tasked with solving the absolute value inequality:

$|2x - 6| < 4$

Step 1: Remove the absolute value

The inequality $|A| < B$ (where $B > 0$) implies two conditions:

$- B < A < B$

Here, $A = 2x - 6$ and $B = 4$, so we can write:

$-4 < 2x - 6 < 4$

Step 2: Solve the compound inequality

Now we solve the inequality $-4 < 2x - 6 < 4$ by isolating $x$.

First, solve the left side of the inequality:

$-4 < 2x - 6$

Add 6 to both sides:

$-4 + 6 < 2x$

$2 < 2x$

Now divide by 2:

$1 < x$

So, $x > 1$.

Next, solve the right side of the inequality:

$2x - 6 < 4$

Add 6 to both sides:

$2x < 10$

Now divide by 2:

$x < 5$

Step 3: Combine the results

From the two parts, we have:

$1 < x < 5$

Final Answer:

The solution to the inequality $|2x - 6| < 4$ is:

$x \in (1, 5)$

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Would you like further details on the solution, or do you have any other questions? Here's a set of related questions to explore:

1. How can you solve an absolute value inequality when the right-hand side is negative?
2. What happens if the inequality is $|2x - 6| \leq 4$ instead of $|2x - 6| < 4$?
3. How can you solve absolute value equations?
4. What are the steps for solving a compound inequality?
5. How would you graph the solution $x \in (1, 5)$ on a number line?

Tip: Always remember that absolute value inequalities can often be rewritten as compound inequalities, making them easier to solve step-by-step!