We are given the inequality:

$\frac{3}{|x+1|} < 5$

To solve this inequality, follow these steps:

Step 1: Isolate the absolute value expression

Multiply both sides by $|x+1|$ to eliminate the denominator. However, since $|x+1|$ is always positive, we don't need to reverse the inequality:

$3 < 5|x+1|$

Step 2: Divide both sides by 5

To simplify:

$\frac{3}{5} < |x+1|$

Step 3: Split into two inequalities

The absolute value inequality $|x+1| > \frac{3}{5}$ implies two cases:

1. $x + 1 > \frac{3}{5}$
2. $x + 1 < -\frac{3}{5}$

Step 4: Solve each case

For Case 1: $x + 1 > \frac{3}{5}$ Subtract 1 from both sides: $x > \frac{3}{5} - 1 = -\frac{2}{5}$

For Case 2: $x + 1 < -\frac{3}{5}$ Subtract 1 from both sides: $x < -\frac{3}{5} - 1 = -\frac{8}{5}$

Step 5: Final solution

Combine both results. The solution is: $x < -\frac{8}{5} \quad \text{or} \quad x > -\frac{2}{5}$ In interval notation, this is: $(-\infty, -\frac{8}{5}) \cup (-\frac{2}{5}, \infty)$

Summary:

The solution is $x \in (-\infty, -\frac{8}{5}) \cup (-\frac{2}{5}, \infty)$.

Would you like more details or have any questions?

Here are 5 related questions you might find useful:

1. How do we solve inequalities with quadratic absolute values?
2. What happens if the inequality had $\leq$ instead of $<$?
3. How do you handle absolute value inequalities where both sides are expressions?
4. Can you explain how absolute value inequalities are visualized on a number line?
5. How do you solve absolute value equations instead of inequalities?

Tip: When dealing with absolute value inequalities, always consider both the positive and negative cases to capture all possible solutions.