The set of real values of ‘x’ satisfying the equality `[(3)/(x)] + [(4)/(x)] = 5` (where [ * ] denotes the greatest integer function) belongs to the interval `(a , (b)/(c ) ]` where `a , b , c in N` and `(b)/( c)` is in its lowest form . Find the value of a + b + c + abc .

Answer

To solve the equation \(\left[\frac{3}{x}\right] + \left[\frac{4}{x}\right] = 5\), where \([\cdot]\) denotes the greatest integer function, we will analyze the values of \(x\) that satisfy this equation step by step.

### Step 1: Understanding the Equation
The equation states that the sum of the greatest integer values of \(\frac{3}{x}\) and \(\frac{4}{x}\) equals 5. We can denote:
- \(a = \left[\frac{3}{x}\right]\)
- \(b = \left[\frac{4}{x}\right]\)

Thus, we have \(a + b = 5\).

### Step 2: Finding Possible Values of \(a\) and \(b\)
Since \(a\) and \(b\) are integers, the possible pairs \((a, b)\) that satisfy \(a + b = 5\) are:
- \((0, 5)\)
- \((1, 4)\)
- \((2, 3)\)
- \((3, 2)\)
- \((4, 1)\)
- \((5, 0)\)

### Step 3: Analyzing Each Case
We will analyze each case to find the corresponding values of \(x\).

**Case 1: \(a = 0, b = 5\)**
- \(\left[\frac{3}{x}\right] = 0 \Rightarrow 0 \leq \frac{3}{x} < 1 \Rightarrow x > 3\)
- \(\left[\frac{4}{x}\right] = 5 \Rightarrow 5 \leq \frac{4}{x} < 6 \Rightarrow \frac{4}{6} < x \leq \frac{4}{5} \Rightarrow \frac{2}{3} < x \leq \frac{4}{5}\)

This case does not yield any valid \(x\) since \(x > 3\) contradicts \(x \leq \frac{4}{5}\).

**Case 2: \(a = 1, b = 4\)**
- \(\left[\frac{3}{x}\right] = 1 \Rightarrow 1 \leq \frac{3}{x} < 2 \Rightarrow \frac{3}{2} < x \leq 3\)
- \(\left[\frac{4}{x}\right] = 4 \Rightarrow 4 \leq \frac{4}{x} < 5 \Rightarrow \frac{4}{5} < x \leq 1\)

This case does not yield any valid \(x\) since \(\frac{3}{2} < x \leq 3\) contradicts \(x \leq 1\).

**Case 3: \(a = 2, b = 3\)**
- \(\left[\frac{3}{x}\right] = 2 \Rightarrow 2 \leq \frac{3}{x} < 3 \Rightarrow 1 < x \leq \frac{3}{2}\)
- \(\left[\frac{4}{x}\right] = 3 \Rightarrow 3 \leq \frac{4}{x} < 4 \Rightarrow 1 < x \leq \frac{4}{3}\)

The valid range for \(x\) is \(1 < x \leq \frac{3}{2}\) and \(1 < x \leq \frac{4}{3}\). The intersection is \(1 < x \leq \frac{4}{3}\).

**Case 4: \(a = 3, b = 2\)**
- \(\left[\frac{3}{x}\right] = 3 \Rightarrow 3 \leq \frac{3}{x} < 4 \Rightarrow \frac{3}{4} < x \leq 1\)
- \(\left[\frac{4}{x}\right] = 2 \Rightarrow 2 \leq \frac{4}{x} < 3 \Rightarrow \frac{4}{3} < x \leq 2\)

This case does not yield any valid \(x\) since \(\frac{3}{4} < x \leq 1\) contradicts \(x \geq \frac{4}{3}\).

**Case 5: \(a = 4, b = 1\)**
- \(\left[\frac{3}{x}\right] = 4 \Rightarrow 4 \leq \frac{3}{x} < 5 \Rightarrow \frac{3}{5} < x \leq \frac{3}{4}\)
- \(\left[\frac{4}{x}\right] = 1 \Rightarrow 1 \leq \frac{4}{x} < 2 \Rightarrow 2 < x \leq 4\)

This case does not yield any valid \(x\) since \(\frac{3}{5} < x \leq \frac{3}{4}\) contradicts \(x \geq 2\).

**Case 6: \(a = 5, b = 0\)**
- \(\left[\frac{3}{x}\right] = 5 \Rightarrow 5 \leq \frac{3}{x} < 6 \Rightarrow \frac{3}{6} < x \leq \frac{3}{5}\)
- \(\left[\frac{4}{x}\right] = 0 \Rightarrow 0 \leq \frac{4}{x} < 1 \Rightarrow x > 4\)

This case does not yield any valid \(x\) since \(x > 4\) contradicts \(x \leq \frac{3}{5}\).

### Step 4: Conclusion
The only valid case is Case 3, which gives us the interval \( (1, \frac{4}{3}] \).

### Step 5: Identifying \(a\), \(b\), and \(c\)
From the interval \( (1, \frac{4}{3}] \):
- \(a = 1\)
- \(b = 4\)
- \(c = 3\)

### Step 6: Calculating \(a + b + c + abc\)
Now, we calculate:
\[ a + b + c + abc = 1 + 4 + 3 + (1 \cdot 4 \cdot 3) = 8 + 12 = 20 \]

### Final Answer
The value of \(a + b + c + abc\) is \(20\).