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The inequality presented is:

\[
|x - 4| < 2 |3x + 1|
\]

Let's solve this step by step.

### Step 1: Analyze the Absolute Values
The inequality contains two absolute value expressions, \( |x - 4| \) and \( |3x + 1| \). We need to consider the cases for these absolute values based on where each expression changes sign.

#### For \( |x - 4| \):
- The expression \( |x - 4| \) changes sign at \( x = 4 \).

#### For \( |3x + 1| \):
- The expression \( |3x + 1| \) changes sign when \( 3x + 1 = 0 \), i.e., at \( x = -\frac{1}{3} \).

These critical points divide the number line into three intervals:
1. \( x < -\frac{1}{3} \)
2. \( -\frac{1}{3} \leq x < 4 \)
3. \( x \geq 4 \)

### Step 2: Solve the Inequality in Each Interval

#### Case 1: \( x < -\frac{1}{3} \)
In this region, both \( x - 4 \) and \( 3x + 1 \) are negative, so:
\[
|x - 4| = -(x - 4) = 4 - x, \quad |3x + 1| = -(3x + 1) = -3x - 1
\]
Substitute these into the inequality:
\[
4 - x < 2(-3x - 1)
\]
Simplify:
\[
4 - x < -6x - 2
\]
Add \( 6x \) to both sides:
\[
5x + 4 < -2
\]
Subtract 4 from both sides:
\[
5x < -6
\]
Divide by 5:
\[
x < -\frac{6}{5}
\]
So, in this case, the solution is \( x < -\frac{6}{5} \).

#### Case 2: \( -\frac{1}{3} \leq x < 4 \)
In this region, \( x - 4 \) is negative, but \( 3x + 1 \) is positive, so:
\[
|x - 4| = -(x - 4) = 4 - x, \quad |3x + 1| = 3x + 1
\]
Substitute these into the inequality:
\[
4 - x < 2(3x + 1)
\]
Simplify:
\[
4 - x < 6x + 2
\]
Add \( x \) to both sides:
\[
4 < 7x + 2
\]
Subtract 2 from both sides:
\[
2 < 7x
\]
Divide by 7:
\[
x > \frac{2}{7}
\]
So, in this case, the solution is \( \frac{2}{7} < x < 4 \).

#### Case 3: \( x \geq 4 \)
In this region, both \( x - 4 \) and \( 3x + 1 \) are positive, so:
\[
|x - 4| = x - 4, \quad |3x + 1| = 3x + 1
\]
Substitute these into the inequality:
\[
x - 4 < 2(3x + 1)
\]
Simplify:
\[
x - 4 < 6x + 2
\]
Subtract \( x \) from both sides:
\[
-4 < 5x + 2
\]
Subtract 2 from both sides:
\[
-6 < 5x
\]
Divide by 5:
\[
x > -\frac{6}{5}
\]
Since we are already in the region \( x \geq 4 \), the solution for this case is simply \( x \geq 4 \).

### Step 3: Combine the Results
From the three cases, we combine the solutions:
- \( x < -\frac{6}{5} \)
- \( \frac{2}{7} < x < 4 \)
- \( x \geq 4 \)

Thus, the full solution is:
\[
x < -\frac{6}{5} \quad \text{or} \quad x > \frac{2}{7}
\]

Would you like further details or clarifications?

Here are some related questions to extend your understanding:
1. How do you solve an absolute value equation?
2. What are the properties of absolute values in inequalities?
3. How can you graph the solution to an absolute value inequality?
4. What are the differences between strict and non-strict inequalities?
5. How would you handle multiple absolute values in an inequality?

**Tip**: Always check the sign changes for absolute values by identifying the critical points where the expressions inside them become zero!

Solve the inequality |x - 4| < 2|3x + 1|.

Explore the step-by-step process to solve the absolute value inequality |x - 4| < 2|3x + 1|. This detailed solution covers key algebraic concepts, including the properties of absolute values and interval analysis. Perfect for students in Grades 10-12.

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