solve symbolically $$ \frac{4}{x+3}\ge0 $$

To solve the inequality 

\[
\frac{4}{x+3} \geq 0,
\]

follow these steps:

### Step 1: Understand the inequality
The inequality will hold when the numerator is non-negative, and the denominator does not cause division by zero.

1. The numerator $ 4 $ is always positive.
2. The denominator $ x + 3 $ will determine the sign of the fraction.

### Step 2: Analyze the denominator
The denominator, $ x + 3 $, affects the sign of the fraction:
- When $ x + 3 > 0 $, the fraction is positive.
- When $ x + 3 = 0 $, the fraction is undefined.
- When $ x + 3 < 0 $, the fraction is negative.

### Step 3: Solve $ x + 3 > 0 $
\[
x + 3 > 0 \implies x > -3.
\]

### Step 4: Write the solution
- For $ x > -3 $, the fraction is positive, so $ \frac{4}{x+3} \geq 0 $ is true.
- For $ x = -3 $, the fraction is undefined.
- For $ x < -3 $, the fraction is negative, so the inequality is false.

### Final Solution:
\[
x \in (-3, \infty).
\]

This is the solution in interval notation. 

---

Would you like a detailed explanation of any step? 

Here are 5 follow-up questions:
1. How does the inequality change if $ \frac{-4}{x+3} \geq 0 $?
2. Can we include $ x = -3 $ in the solution?
3. What happens if the numerator also depends on $ x $?
4. How would this inequality change if the fraction were $ \leq 0 $ instead?
5. What are some applications of such inequalities in real-life scenarios?

**Tip:** Always check for division by zero when solving rational inequalities.

Solve symbolically $$ \frac{4}{x+3} \geq 0 $$

Discover how to solve the rational inequality 4/(x+3) ≥ 0 step-by-step. Learn about inequalities, rational expressions, and interval notation, complete with a detailed solution for Grades 10-12 students.

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