Any equation or inequality with variables in it is a predicate in the domain of real numbers. For the following statement, tell whether the statement is true or false.
(∀*x*)(*x*3 > *x*)

The statement given is:

\[
\forall x (x^3 > x)
\]

This means "for all real numbers \( x \), \( x^3 \) is greater than \( x \)." To determine whether this is true or false, we need to analyze the inequality \( x^3 > x \) for different values of \( x \) over the real numbers.

1. **For \( x = 1 \):** 
   \[
   1^3 = 1 \quad \text{and} \quad 1 = 1
   \]
   The inequality is not true because \( 1^3 \) is equal to \( 1 \), not greater.

2. **For \( x = 0 \):** 
   \[
   0^3 = 0 \quad \text{and} \quad 0 = 0
   \]
   The inequality is also not true here because \( 0^3 \) is equal to \( 0 \).

3. **For \( x = -1 \):** 
   \[
   (-1)^3 = -1 \quad \text{and} \quad -1 = -1
   \]
   Again, the inequality is not true, as \( (-1)^3 \) equals \( -1 \), not greater than \( -1 \).

4. **For values of \( x \) between 0 and 1 (e.g., \( x = 0.5 \)):**
   \[
   (0.5)^3 = 0.125 \quad \text{and} \quad 0.125 < 0.5
   \]
   Here the inequality does not hold either, since \( 0.5^3 \) is less than \( 0.5 \).

5. **For values of \( x \) less than -1 (e.g., \( x = -2 \)):**
   \[
   (-2)^3 = -8 \quad \text{and} \quad -8 < -2
   \]
   The inequality does not hold here either.

Therefore, the statement \( \forall x (x^3 > x) \) is **false** because the inequality does not hold for all real numbers \( x \). It only holds for values of \( x \) greater than 1 or less than -1.

Would you like more details or have any questions?

### Related Questions:
1. What are other examples of predicates in mathematics?
2. How would you prove whether an inequality holds for all real numbers?
3. Can you visualize the inequality \( x^3 > x \) on a graph?
4. How would this statement change if we considered a different domain, such as integers?
5. What methods can we use to test inequalities involving polynomials?

**Tip:** When testing inequalities, it's useful to check values around key points like 0, 1, and -1 to better understand the behavior of the inequality.

Any equation or inequality with variables in it is a predicate in the domain of real numbers. For the following statement, tell whether the statement is true or false. (∀x)(x^3 > x)

Determine whether the statement (∀x)(x^3 > x) is true or false by analyzing the inequality x^3 > x over the real numbers. Understand key concepts in algebra, including predicates and inequalities, suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation