Let's solve the inequality $\frac{x^2 + 3x - 2}{x^2 + x + 3} < 1$.

Step 1: Subtract 1 from both sides to set the inequality to zero:

$\frac{x^2 + 3x - 2}{x^2 + x + 3} - 1 < 0$

Step 2: Combine the terms under a common denominator:

$\frac{x^2 + 3x - 2}{x^2 + x + 3} - \frac{x^2 + x + 3}{x^2 + x + 3} < 0$

$\frac{(x^2 + 3x - 2) - (x^2 + x + 3)}{x^2 + x + 3} < 0$

Step 3: Simplify the numerator:

$(x^2 + 3x - 2) - (x^2 + x + 3) = x^2 + 3x - 2 - x^2 - x - 3 = 2x - 5$

So the inequality becomes:

$\frac{2x - 5}{x^2 + x + 3} < 0$

Step 4: Analyze the inequality

We need to determine when $\frac{2x - 5}{x^2 + x + 3} < 0$.

The numerator:

The numerator, $2x - 5$, changes sign when $2x - 5 = 0$, i.e., at $x = \frac{5}{2}$.

• $2x - 5 > 0$ when $x > \frac{5}{2}$
• $2x - 5 < 0$ when $x < \frac{5}{2}$

The denominator:

The denominator, $x^2 + x + 3$, is always positive because its discriminant $1^2 - 4(1)(3) = 1 - 12 = -11$ is negative, meaning it has no real roots and never changes sign.

Step 5: Solve the inequality

Since the denominator is always positive, the inequality will hold wherever the numerator $2x - 5$ is negative. This occurs when $x < \frac{5}{2}$.

Final Solution:

The solution to the inequality is:

$x < \frac{5}{2}$

Would you like further details on any of these steps?

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Here are some related questions to consider:

1. How do you solve inequalities with rational expressions?
2. What are the properties of quadratic discriminants?
3. How can you analyze the behavior of polynomials based on their degree?
4. How do you solve inequalities involving more complex polynomials?
5. What happens if both the numerator and denominator change signs in an inequality?

Tip: When dealing with rational inequalities, always check both the numerator and denominator for critical points to understand the sign changes.