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!