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We are asked to solve the inequality \( |1.3x + 7.8| > 0 \). This involves dealing with the absolute value function, which has two cases for inequality:

### Step 1: Break into two cases
Since the absolute value expression is greater than 0, we can split the inequality as follows:

1. \( 1.3x + 7.8 > 0 \)
2. \( 1.3x + 7.8 < 0 \)

### Case 1: \( 1.3x + 7.8 > 0 \)

Solve for \( x \):

\[
1.3x > -7.8
\]

Now divide both sides by \( 1.3 \):

\[
x > \frac{-7.8}{1.3} = -6
\]

So, \( x > -6 \).

### Case 2: \( 1.3x + 7.8 < 0 \)

In this case, the absolute value would be positive, but not zero. However, the inequality only requires the values where \( 1.3x + 7.8 \neq 0 \). For absolute value inequalities, \( |A| > 0 \) is satisfied for all \( A \neq 0 \).

Thus, the solution is:

\[
x \neq -6
\]

### Conclusion:
The final solution is \( x \neq -6 \), meaning that the absolute value will always be positive except when \( x = -6 \).

Would you like further details on any step, or do you have questions?

Here are five related questions:
1. What is the general method for solving absolute value inequalities?
2. How would you solve \( |ax + b| < c \) for a different inequality?
3. How does this inequality compare to quadratic inequalities?
4. What happens when the right-hand side of an absolute value inequality is negative?
5. Can you solve absolute value inequalities graphically?

**Tip**: Always check if an absolute value inequality is set against a positive or negative number, as this will determine if there's a valid solution.

This webpage provides a step-by-step solution to the absolute value inequality |1.3x + 7.8| > 0. Learn the breakdown of the solution process using absolute value properties. Suitable for students in Grades 9-11.

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