The inequality given in the image is:

$|5x + \frac{1}{2} - 4| < 8$

We need to solve this absolute value inequality step by step.

Step 1: Simplify the expression inside the absolute value.

$5x + \frac{1}{2} - 4$

Convert $-4$ into a fraction with denominator 2:

$5x + \frac{1}{2} - \frac{8}{2} = 5x - \frac{7}{2}$

So, the inequality becomes:

$|5x - \frac{7}{2}| < 8$

Step 2: Remove the absolute value by writing two inequalities.

When we solve absolute value inequalities, we have two cases:

1. $5x - \frac{7}{2} < 8$
2. $-(5x - \frac{7}{2}) < 8$ (or equivalently $5x - \frac{7}{2} > -8$)

Step 3: Solve each inequality separately.

Case 1: Solve $5x - \frac{7}{2} < 8$

Add $\frac{7}{2}$ to both sides:

$5x < 8 + \frac{7}{2}$

Convert 8 into a fraction:

$5x < \frac{16}{2} + \frac{7}{2} = \frac{23}{2}$

Now, divide both sides by 5:

$x < \frac{23}{10}$

Case 2: Solve $5x - \frac{7}{2} > -8$

Add $\frac{7}{2}$ to both sides:

$5x > -8 + \frac{7}{2}$

Convert $-8$ into a fraction:

$5x > -\frac{16}{2} + \frac{7}{2} = -\frac{9}{2}$

Now, divide both sides by 5:

$x > -\frac{9}{10}$

Step 4: Combine the results.

The solution to the inequality is:

$-\frac{9}{10} < x < \frac{23}{10}$

Final Answer:

The solution set is:

$(-\frac{9}{10}, \frac{23}{10})$

Thus, the correct choice is A, and the solution set in interval notation is $\left( -\frac{9}{10}, \frac{23}{10} \right)$.

Would you like more details or have any questions?

Here are 5 follow-up questions:

1. How do you handle absolute value inequalities that involve "greater than" instead of "less than"?
2. What happens if you add or subtract a constant inside the absolute value?
3. How can you represent the solution to inequalities using a number line?
4. What is the difference between strict inequalities (like "<") and non-strict ones (like "≤")?
5. How does multiplying or dividing by a negative number affect inequalities?

Tip: When solving inequalities, be careful with signs, especially when multiplying or dividing by negative numbers, as this reverses the inequality.