If x is a rational number then x squared greater then or equal to x

To determine if \( x^2 \geq x \) holds true for all rational numbers \( x \), we can analyze the inequality:

### Start with the inequality:
\[
x^2 \geq x
\]

### Rearrange the inequality:
\[
x^2 - x \geq 0
\]

### Factor the expression:
\[
x(x - 1) \geq 0
\]

This inequality holds when the product \( x(x - 1) \) is non-negative.

### Analyze the factors:
1. **\( x > 1 \):** Both \( x \) and \( x - 1 \) are positive, so \( x(x - 1) \geq 0 \).
2. **\( x = 1 \):** The product is zero, so \( x(x - 1) = 0 \).
3. **\( 0 < x < 1 \):** \( x \) is positive, but \( x - 1 \) is negative, so \( x(x - 1) < 0 \).
4. **\( x = 0 \):** The product is zero, so \( x(x - 1) = 0 \).
5. **\( x < 0 \):** Both \( x \) and \( x - 1 \) are negative, so \( x(x - 1) \geq 0 \).

### Conclusion:
The inequality \( x^2 \geq x \) holds for all rational numbers except when \( 0 < x < 1 \). In this interval, \( x^2 < x \).

Would you like more details or have any questions?

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Here are 5 related questions:

1. What happens if \( x \) is an irrational number? Does the inequality \( x^2 \geq x \) still hold?
2. How does this analysis change if we consider \( x^2 \leq x \) instead?
3. Can you find specific values of \( x \) where \( x^2 = x \)?
4. What would happen if we instead considered \( x^3 \geq x \)? How would the solution change?
5. How does this inequality behave for complex numbers?

**Tip:** When factoring inequalities, always consider the sign of each factor to determine where the inequality holds.

Explore the inequality x^2 ≥ x for rational numbers. Learn how to analyze and solve this inequality, including detailed explanations and examples. Suitable for high school students studying algebra and inequalities.

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