In equation no.4 where does -2x -3/x-3 came from?

Apologies for the confusion in the explanation for equation no. 4. Let’s go through it step-by-step to clarify how to solve the inequality correctly:

### 4. \( \frac{3}{x + 3} \geq 2 \)

To solve this inequality, let's follow the proper steps:

1. **Start with the given inequality:**

   \[
   \frac{3}{x + 3} \geq 2
   \]

2. **Subtract 2 from both sides to set up for a common denominator:**

   \[
   \frac{3}{x + 3} - 2 \geq 0
   \]

3. **Rewrite the inequality with a common denominator:**

   \[
   \frac{3 - 2(x + 3)}{x + 3} \geq 0
   \]

   Here, \(2\) can be written as \(\frac{2(x + 3)}{x + 3}\) to have the same denominator as \(\frac{3}{x + 3}\).

4. **Expand and simplify the numerator:**

   \[
   \frac{3 - 2(x + 3)}{x + 3} \geq 0
   \]

   Simplify the numerator:

   \[
   3 - 2(x + 3) = 3 - 2x - 6 = -2x - 3
   \]

   Thus, the expression becomes:

   \[
   \frac{-2x - 3}{x + 3} \geq 0
   \]

This is where \(\frac{-2x - 3}{x + 3}\) comes from: simplifying the numerator after rewriting the inequality to have a common denominator.

5. **Find the critical points:**

   The critical points are found where the numerator and denominator equal zero:

   - **Numerator:** \(-2x - 3 = 0 \Rightarrow x = -\frac{3}{2}\)
   - **Denominator:** \(x + 3 = 0 \Rightarrow x = -3\)

6. **Test the intervals around these critical points:**

   - **Interval 1:** \(x < -3\)  
     Both the numerator \((-2x - 3)\) and the denominator \((x + 3)\) are negative, so the fraction is positive.
   
   - **Interval 2:** \(-3 < x < -\frac{3}{2}\)  
     The numerator is positive and the denominator is negative, making the fraction negative.

   - **Interval 3:** \(x > -\frac{3}{2}\)  
     Both the numerator and denominator are positive, making the fraction positive.

7. **Determine where the inequality holds:**

   The inequality \(\frac{-2x - 3}{x + 3} \geq 0\) is satisfied when the fraction is positive or zero:

   - When \(x \leq -3\), the fraction is positive.
   - When \(x \geq -\frac{3}{2}\), the fraction is positive or zero.

Thus, the solution to the inequality is:

\[
x \leq -3 \quad \text{or} \quad x \geq -\frac{3}{2}
\]

### Summary

The expression \(\frac{-2x - 3}{x + 3}\) came from subtracting \(2\) from \(\frac{3}{x + 3}\) and rewriting it as a single fraction with a common denominator, then simplifying the numerator. 

If you have any more questions or need further clarification, feel free to ask!

### Further Questions:

1. Why is it necessary to rewrite the equation with a common denominator when solving rational inequalities?
2. What are some general strategies for simplifying complex rational expressions?
3. How can you determine the sign of an interval when solving inequalities involving fractions?
4. What happens to the inequality if the numerator or denominator is set to zero?
5. How would the solution change if the inequality sign were reversed (e.g., \(\leq\) instead of \(\geq\))?

### Tip:

When solving inequalities involving fractions, always pay attention to the points where the denominator is zero. These points are important because the expression is undefined there, and they help in determining the sign of the expression over different intervals.

Explore the step-by-step solution to the rational inequality \( \frac{3}{x + 3} \geq 2 \), including explanations on common denominators and simplifying rational expressions. Suitable for high school students studying inequalities in algebra.

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