We are tasked with solving the modular equation:

\[
6x \equiv 4 \ (\text{mod} \ 9)
\]

This means we want to find an integer \( x \) such that when we multiply \( x \) by 6, the result is congruent to 4 modulo 9. The process to solve this involves finding the multiplicative inverse of 6 modulo 9 and then multiplying both sides of the equation by that inverse.

### Step 1: Check if an inverse exists
The multiplicative inverse of \( 6 \) modulo \( 9 \) exists if and only if \( \gcd(6, 9) = 1 \), where \( \gcd \) is the greatest common divisor. In this case:

\[
\gcd(6, 9) = 3
\]

Since the GCD is not 1, there is no multiplicative inverse of 6 modulo 9. This means the equation does not have a unique solution modulo 9, but it may still have solutions depending on other factors.

### Step 2: Simplify and check for possible solutions
Since we know there's no inverse, we can attempt to solve the equation by trying values of \( x \) directly to see if any work. We'll substitute values of \( x \) and compute \( 6x \mod 9 \) to find where it's congruent to 4.

- \( x = 0 \): \( 6(0) \mod 9 = 0 \)
- \( x = 1 \): \( 6(1) \mod 9 = 6 \)
- \( x = 2 \): \( 6(2) \mod 9 = 3 \)
- \( x = 3 \): \( 6(3) \mod 9 = 0 \)
- \( x = 4 \): \( 6(4) \mod 9 = 6 \)
- \( x = 5 \): \( 6(5) \mod 9 = 3 \)
- \( x = 6 \): \( 6(6) \mod 9 = 0 \)

As we can see, none of these values satisfy the equation \( 6x \equiv 4 \ (\text{mod} \ 9) \).

### Conclusion:
The equation \( 6x \equiv 4 \ (\text{mod} \ 9) \) has no solution in the integers, because the greatest common divisor of 6 and 9 is 3, which divides 6 but not 4. Therefore, no integer \( x \) satisfies this equation.

Would you like to see more details or ask any questions about this solution?

Here are 5 follow-up questions for deeper understanding:
1. How do we find the multiplicative inverse in modular arithmetic when it exists?
2. Can you explain more about the significance of the greatest common divisor in modular equations?
3. What happens when the GCD is 1, and how does that affect solutions?
4. Could you provide an example where an equation has multiple solutions in modular arithmetic?
5. How can we generalize solving equations in modulo arithmetic when there's no inverse?

**Tip**: Always check the greatest common divisor between the coefficient and the modulus in modular equations—it determines whether a solution exists and whether the equation is solvable!

Discover how to solve the modular equation 6x ≡ 4 (mod 9). This problem covers essential modular arithmetic concepts like greatest common divisors and inverse existence, providing a step-by-step solution. Suitable for high school students in Grades 10-12.

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