To solve the equation $5[x] = 3 + 3x$, where $[x]$ denotes the Greatest Integer Function (also called the floor function), we need to break down the steps carefully.

Step-by-Step Solution:

The Greatest Integer Function $[x]$ returns the greatest integer less than or equal to $x$. So, we are tasked with solving an equation involving a floor function.

1. Start by analyzing the equation: $5[x] = 3 + 3x$ where $[x]$ is the greatest integer less than or equal to $x$.

2. Rewrite the equation: $[x] = n \quad \text{where} \quad n \in \mathbb{Z}$ means $n$ is an integer, and $n \leq x < n + 1$.

   So, we can rewrite the equation as: $5n = 3 + 3x$ which simplifies to: $5n - 3 = 3x$ $x = \frac{5n - 3}{3}$

3. Determine the possible values for $n$: Since $n \leq x < n+1$, substitute $x = \frac{5n - 3}{3}$ into this range: $n \leq \frac{5n - 3}{3} < n + 1$

   Let's solve these two inequalities one by one.

First Inequality:

$n \leq \frac{5n - 3}{3}$ Multiply through by 3 to eliminate the denominator: $3n \leq 5n - 3$ $3n - 5n \leq -3$ $-2n \leq -3$ $n \geq \frac{3}{2}$ Since $n$ is an integer, the smallest integer $n$ can be is $2$.

Second Inequality:

$\frac{5n - 3}{3} < n + 1$ Multiply through by 3: $5n - 3 < 3n + 3$ $5n - 3n < 3 + 3$ $2n < 6$ $n < 3$ Thus, $n$ can be at most 2.

Conclusion:

From the inequalities, $n$ must be $2$. Substituting $n = 2$ into the expression for $x$:

$x = \frac{5(2) - 3}{3} = \frac{10 - 3}{3} = \frac{7}{3} \approx 2.33$

Since $n = 2$, we have $2 \leq x < 3$, which is valid because $\frac{7}{3}$ is within this range.

Thus, the solution to the equation is: $x = \frac{7}{3}$

---

Would you like further details on this process? Feel free to ask any questions!

Related Questions:

1. What is the definition and behavior of the Greatest Integer Function?
2. How can we solve equations involving the Greatest Integer Function step by step?
3. How does the Greatest Integer Function behave with negative numbers?
4. Can the solution for $x$ change if the right-hand side of the equation was altered (e.g., changing constants)?
5. How do we handle fractional parts when working with floor functions in equations?

Tip:

When solving floor function problems, always check if the value of $x$ obtained falls within the range defined by the floor function.