To solve the inequality $4 | -x + 4 | < 12$, let's follow these steps:

Step 1: Isolate the Absolute Value

First, divide both sides by 4: $| -x + 4 | < 3$

Step 2: Set Up Two Cases

An absolute value inequality $|A| < B$ implies $-B < A < B$. Here, we have: $-3 < -x + 4 < 3$

Step 3: Solve the Compound Inequality

Part 1: $-3 < -x + 4$

Subtract 4 from both sides: $-7 < -x$ Now, divide by -1 (remember to reverse the inequality sign): $x < 7$

Part 2: $-x + 4 < 3$

Subtract 4 from both sides: $-x < -1$ Then, divide by -1 (reverse the inequality sign again): $x > 1$

Step 4: Combine the Results

The solution is: $1 < x < 7$ Or, in interval notation: $(1, 7)$

Final Answer

$x \in (1, 7)$

Would you like more details or have any questions?

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Related Practice Questions:

1. Solve $|2x - 5| < 7$.
2. Solve $3 |x + 4| \geq 15$.
3. Solve $|5x + 3| \leq 8$.
4. Solve $2 |x - 1| + 3 < 11$.
5. Solve $|3x - 2| > 4$.

Tip: When solving absolute value inequalities, remember to split the inequality into two cases and check for direction changes when dividing by negative values.