the maximum integral value of x satisfying the inequality is (x^2-x+1)/(x^2+x+1)(4-x^2) >0



We are tasked with finding the maximum integral value of \( x \) that satisfies the inequality:

\[
\frac{(x^2 - x + 1)}{(x^2 + x + 1)}(4 - x^2) > 0
\]

### Step 1: Analyzing the factors
The inequality can be split into two main factors:

1. \( \frac{x^2 - x + 1}{x^2 + x + 1} \)
2. \( 4 - x^2 \)

We'll analyze each factor separately.

### Step 2: Understanding the quadratic expressions

- \( x^2 - x + 1 \) and \( x^2 + x + 1 \) are quadratic expressions.
  - \( x^2 - x + 1 \) is always positive for all \( x \), since its discriminant \( (-1)^2 - 4 \cdot 1 \cdot 1 = -3 \) is negative, meaning it has no real roots and opens upwards (its minimum value is positive).
  - \( x^2 + x + 1 \) is also always positive for all \( x \) because its discriminant \( (1)^2 - 4 \cdot 1 \cdot 1 = -3 \) is negative, meaning no real roots and the expression is always positive.

Thus, the fraction \( \frac{x^2 - x + 1}{x^2 + x + 1} \) is always positive for all \( x \).

### Step 3: Analyzing \( 4 - x^2 \)

- The expression \( 4 - x^2 \) represents a parabola that opens downwards. It is positive when \( x^2 < 4 \), i.e., when \( -2 < x < 2 \), and negative when \( x^2 > 4 \), i.e., for \( x < -2 \) or \( x > 2 \).

### Step 4: Overall inequality

For the product of these factors to be greater than 0, the second factor \( 4 - x^2 \) must be positive, as the first factor is always positive. Therefore, we need:

\[
4 - x^2 > 0 \quad \text{or} \quad -2 < x < 2
\]

### Step 5: Finding the maximum integral value

The integral values of \( x \) that satisfy \( -2 < x < 2 \) are \( x = -1, 0, 1 \). The maximum integral value is:

\[
\boxed{1}
\]

### Do you want more details or have any questions?

### Related Questions:
1. How do discriminants help in understanding quadratic expressions?
2. What does it mean for a quadratic function to always be positive?
3. How does the range of a quadratic function affect inequalities?
4. Can you show an example of solving a similar inequality with different quadratics?
5. How does changing the sign of a term affect the solution of an inequality?

### Tip:
Always check the sign of each factor in an inequality separately before combining them to solve.

The maximum integral value of x satisfying the inequality is (x^2 - x + 1) / (x^2 + x + 1) (4 - x^2) > 0

Learn how to find the maximum integral value of x that satisfies the inequality (x^2 - x + 1) / (x^2 + x + 1)(4 - x^2) > 0. This problem involves analyzing quadratic expressions, discriminants, and factoring inequalities. Suitable for high school students in Grades 10-12.

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