The given inequality is:

$|x| + 3|x| \leq 7$

Let's simplify and solve this step by step.

Step 1: Combine the absolute value terms

We can combine the two terms involving $|x|$:

$|x| + 3|x| = 4|x|$

Thus, the inequality becomes:

$4|x| \leq 7$

Step 2: Solve for $|x|$

Divide both sides of the inequality by 4:

$|x| \leq \frac{7}{4}$

Step 3: Remove the absolute value

Since $|x| \leq \frac{7}{4}$, this means:

$-\frac{7}{4} \leq x \leq \frac{7}{4}$

Final Solution:

The solution to the inequality is:

$x \in \left[-\frac{7}{4}, \frac{7}{4}\right]$

Would you like more details, or do you have any questions about this? Here are some related questions that could further your understanding:

1. What is an absolute value, and how does it affect inequalities?
2. How do you solve inequalities involving multiple absolute value expressions?
3. Can you explain how to graph the solution on a number line?
4. What happens if the inequality was strict, i.e., $4|x| < 7$?
5. How would the solution change if the right-hand side of the inequality were negative?

Tip: Always isolate the absolute value expression first when solving absolute value inequalities.