The inequality provided is:

$|x - 4| \leq 2(3x + 1)$

To solve this, we first analyze the inequality by considering the absolute value expression, which defines two conditions based on the term inside the absolute value being non-negative or negative. Here's how to proceed:

1. When $x - 4 \geq 0$ (i.e., $x \geq 4$):

   • The inequality simplifies to: $x - 4 \leq 6x + 2$
   • Rearranging gives: $-4 - 2 \leq 6x - x$ $-6 \leq 5x$ $x \geq -\frac{6}{5}$

   Since $x$ must also be greater than or equal to 4, the solution in this case is: $x \geq 4$

2. When $x - 4 < 0$ (i.e., $x < 4$):

   • The inequality simplifies to: $-(x - 4) \leq 6x + 2$ $-x + 4 \leq 6x + 2$
   • Rearranging gives: $4 - 2 \leq 6x + x$ $2 \leq 7x$ $x \geq \frac{2}{7}$

   Since $x$ must also be less than 4, the solution in this case is: $\frac{2}{7} \leq x < 4$

Combining both cases, the complete solution to the inequality is:

$\frac{2}{7} \leq x$

Here, the critical points to check are $x = 4$ and $x = \frac{2}{7}$, and $x = -\frac{6}{5}$. Since $-\frac{6}{5}$ does not affect the solution due to $x \geq \frac{2}{7}$, the final interval of solutions is correct.

Would you like more details or explanations on any part of the solution?

Related Questions:

1. What is the graphical representation of the solution?
2. How would the inequality change if it was strict $|x - 4| < 2(3x + 1)$?
3. What is the importance of checking boundary values in inequalities involving absolute values?
4. How do the solutions change if the coefficients in the inequality are altered?
5. Could there be a situation where no solution exists for a similar type of inequality?

Tip:

Always remember to consider both cases in absolute value inequalities: one where the expression inside the absolute value is non-negative and the other where it is negative.