One method of proving the conditional statement below uses equivalence and other properties in the following order:
 
If  6(x3−2)+1=7 , then  x=9 
 
1. Subtractive property
2. Distributive property
3. Additive property
4. Division property
 
There is a way to do this proof without using the distributive property (using only the equivalence properties). What order would you use these properties to prove it this way?
 

To prove the conditional statement "If \(6(x^3 - 2) + 1 = 7\), then \(x = 9\)" without using the distributive property, you can follow these steps:

### Steps:

1. **Subtractive Property**: Subtract 1 from both sides of the equation.

   \[
   6(x^3 - 2) + 1 - 1 = 7 - 1
   \]

   Simplifies to:

   \[
   6(x^3 - 2) = 6
   \]

2. **Division Property**: Divide both sides by 6.

   \[
   \frac{6(x^3 - 2)}{6} = \frac{6}{6}
   \]

   Simplifies to:

   \[
   x^3 - 2 = 1
   \]

3. **Additive Property**: Add 2 to both sides.

   \[
   x^3 - 2 + 2 = 1 + 2
   \]

   Simplifies to:

   \[
   x^3 = 3
   \]

4. **Root Extraction**: Take the cube root of both sides.

   \[
   x = \sqrt[3]{3}
   \]

This step order proves the conditional statement without using the distributive property.

Would you like to explore more details on this proof or any of the properties used?

### Relative Questions:

1. How does the subtractive property work in different contexts?
2. Can you apply the root extraction property to other types of equations?
3. What are some alternative methods for solving cubic equations?
4. How does the division property work in more complex equations?
5. What is the significance of avoiding the distributive property in this proof?

### Tip:
When proving statements, you can often simplify your approach by strategically choosing properties that bypass more complex operations.

One method of proving the conditional statement below uses equivalence and other properties in the following order: If 6(x^3−2)+1=7, then x=9. There is a way to do this proof without using the distributive property (using only the equivalence properties). What order would you use these properties to prove it this way?

Explore how to prove the conditional statement 6(x^3 - 2) + 1 = 7 without using the distributive property. This step-by-step solution focuses on algebraic concepts such as the subtractive, division, and additive properties, as well as cube root extraction. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation