solve the rational inequalities x²+3x+8/x²+4x+6<0

The inequality given is:

\[
\frac{x^2 + 3x + 8}{x^2 + 4x + 6} < 0
\]

To solve this rational inequality, follow these steps:

### Step 1: Analyze the numerator and denominator
We need to examine the signs of the numerator and denominator.

#### Numerator: \( x^2 + 3x + 8 \)
The discriminant of the quadratic \( x^2 + 3x + 8 \) is:

\[
\Delta = 3^2 - 4(1)(8) = 9 - 32 = -23
\]

Since the discriminant is negative, the quadratic has no real roots and is always positive (as \( x^2 + 3x + 8 \) opens upwards). Hence, the numerator is always positive for all real values of \(x\).

#### Denominator: \( x^2 + 4x + 6 \)
The discriminant of the quadratic \( x^2 + 4x + 6 \) is:

\[
\Delta = 4^2 - 4(1)(6) = 16 - 24 = -8
\]

Since the discriminant is also negative, the quadratic \( x^2 + 4x + 6 \) has no real roots and is always positive (since it also opens upwards). Therefore, the denominator is always positive for all real values of \(x\).

### Step 2: Determine the sign of the rational expression
Since both the numerator and the denominator are always positive for all \(x\), the expression:

\[
\frac{x^2 + 3x + 8}{x^2 + 4x + 6}
\]

is always positive for all real values of \(x\). Therefore, it is impossible for this expression to be less than zero.

### Conclusion:
There are no solutions to the inequality:

\[
\frac{x^2 + 3x + 8}{x^2 + 4x + 6} < 0
\]

since the expression is always positive.

Would you like more details or have any further questions?

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Here are five related questions for deeper understanding:
1. How can you determine whether a quadratic expression is always positive or negative?
2. What is the significance of the discriminant of a quadratic equation?
3. Can a rational expression be less than zero if both the numerator and denominator are positive?
4. How would the solution change if the inequality were \( \geq 0 \) instead of \( < 0 \)?
5. How can inequalities involving rational expressions be applied in real-world scenarios?

**Tip**: Always check the discriminant to determine whether a quadratic has real roots, which helps in determining the sign of the expression.

solve the rational inequalities x² + 3x + 8 / x² + 4x + 6 < 0

Explore how to solve the rational inequality x² + 3x + 8 / x² + 4x + 6 < 0 using discriminant analysis. Learn why both the numerator and denominator are always positive, resulting in no solutions. Ideal for students in Grades 10-12, focusing on quadratic equations and rational inequalities.

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