The given absolute value inequality is:

$|14x| \leq 7$

To solve this, we need to consider the definition of absolute value. The inequality $|A| \leq B$ can be rewritten as:

$-B \leq A \leq B$

In this case, $A = 14x$ and $B = 7$, so we have:

$-7 \leq 14x \leq 7$

Next, we solve for $x$ by dividing all parts of the inequality by 14:

$\frac{-7}{14} \leq \frac{14x}{14} \leq \frac{7}{14}$

Simplifying this, we get:

$-\frac{1}{2} \leq x \leq \frac{1}{2}$

So, the solution to the inequality is:

$x \in \left[ -\frac{1}{2}, \frac{1}{2} \right]$

This means that $x$ can be any value between $-\frac{1}{2}$ and $\frac{1}{2}$, inclusive.

Would you like any further details or have any questions?

Here are some related questions:

1. How do you solve absolute value inequalities involving "greater than"?
2. What is the difference between solving $|x| \leq a$ and $|x| \geq a$?
3. How do you graph the solution to an absolute value inequality on a number line?
4. Can absolute value inequalities have no solution? If so, when?
5. How would the inequality change if it was $|14x| > 7$ instead?

Tip: When solving absolute value inequalities, remember that you often need to split the problem into two cases to remove the absolute value.