To solve the inequality \( 10 \lvert -5n \rvert + 8 < 50 \), follow these steps:

### Step 1: Isolate the absolute value term
Subtract 8 from both sides:

\[
10 \lvert -5n \rvert < 50 - 8
\]
\[
10 \lvert -5n \rvert < 42
\]

### Step 2: Divide by 10
To further simplify, divide both sides by 10:

\[
\lvert -5n \rvert < \frac{42}{10}
\]
\[
\lvert -5n \rvert < 4.2
\]

### Step 3: Remove the absolute value
Since \( \lvert -5n \rvert \) represents both positive and negative values of \( -5n \), rewrite the inequality as two separate inequalities:

\[
-4.2 < -5n < 4.2
\]

### Step 4: Solve for \( n \)
Now, divide the entire inequality by \( -5 \). Since we are dividing by a negative number, reverse the inequality signs:

\[
\frac{-4.2}{-5} > n > \frac{4.2}{-5}
\]
\[
0.84 > n > -0.84
\]

### Final answer:
The solution is:

\[
-0.84 < n < 0.84
\]

### Summary:
The values of \( n \) that satisfy the inequality are in the range \( -0.84 < n < 0.84 \).

Would you like more details or have any questions?

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Here are five related questions:
1. How do you solve inequalities with absolute values?
2. What happens to the inequality when dividing by a negative number?
3. How do you interpret compound inequalities?
4. Can you graph the solution set on a number line?
5. What are other methods to solve absolute value inequalities?

**Tip:** Always reverse inequality signs when multiplying or dividing by a negative number.

Explore the solution to the absolute value inequality 10|-5n|+8<50, with a detailed, step-by-step breakdown. This algebra problem introduces key concepts such as absolute value inequalities and how to handle inequalities when dividing by negative numbers. Ideal for students in Grades 9-11.

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