Find the number of positive integral value of `x` satisfying the inequality `((3^(x)-5^(x))(x-2))/((x^(2)+5x+2))ge0`

Answer

To solve the inequality \(\frac{(3^x - 5^x)(x - 2)}{(x^2 + 5x + 2)} \geq 0\), we will analyze the expression step by step.

### Step 1: Identify the critical points

The expression is a fraction, and we need to find the points where the numerator and denominator are zero.

1. **Numerator**: \(3^x - 5^x = 0\)
- This occurs when \(3^x = 5^x\).
- Taking logarithm on both sides, we get \(x \log(3) = x \log(5)\).
- This implies \(x = 0\) (since \(\log(3) \neq \log(5)\)).

2. **Numerator**: \(x - 2 = 0\)
- This occurs when \(x = 2\).

3. **Denominator**: \(x^2 + 5x + 2 = 0\)
- We can use the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{-5 \pm \sqrt{25 - 8}}{2} = \frac{-5 \pm \sqrt{17}}{2}. \]
- The roots are not integers, but we will note them for sign analysis.

### Step 2: Analyze the sign of the expression

We need to analyze the sign of the expression in the intervals determined by the critical points: \((-\infty, 0)\), \((0, 2)\), and \((2, +\infty)\).

1. **Interval \((-\infty, 0)\)**:
- Choose \(x = -1\):
\[ 3^{-1} - 5^{-1} < 0, \quad -1 - 2 < 0, \quad (-1)^2 + 5(-1) + 2 < 0. \]
- The numerator is negative, and the denominator is positive. Thus, the expression is negative.

2. **Interval \((0, 2)\)**:
- Choose \(x = 1\):
\[ 3^1 - 5^1 < 0, \quad 1 - 2 < 0, \quad 1^2 + 5(1) + 2 > 0. \]
- The numerator is negative, and the denominator is positive. Thus, the expression is negative.

3. **Interval \((2, +\infty)\)**:
- Choose \(x = 3\):
\[ 3^3 - 5^3 > 0, \quad 3 - 2 > 0, \quad 3^2 + 5(3) + 2 > 0. \]
- The numerator is positive, and the denominator is positive. Thus, the expression is positive.

### Step 3: Evaluate the endpoints

- At \(x = 0\):
\[ \frac{(3^0 - 5^0)(0 - 2)}{(0^2 + 5(0) + 2)} = \frac{(1 - 1)(-2)}{2} = 0. \]
- The expression equals zero.

- At \(x = 2\):
\[ \frac{(3^2 - 5^2)(2 - 2)}{(2^2 + 5(2) + 2)} = \frac{(9 - 25)(0)}{(4 + 10 + 2)} = 0. \]
- The expression equals zero.

### Step 4: Summary of intervals

The expression is:
- Negative in \((-\infty, 0)\)
- Negative in \((0, 2)\)
- Zero at \(x = 0\) and \(x = 2\)
- Positive in \((2, +\infty)\)

### Step 5: Identify positive integral solutions

The positive integral values of \(x\) that satisfy the inequality \(\frac{(3^x - 5^x)(x - 2)}{(x^2 + 5x + 2)} \geq 0\) are:
- \(x = 2\) (where the expression equals zero)
- \(x = 3\) (where the expression is positive)
- \(x = 4\) (where the expression is positive)
- Continuing this way, all integers greater than 2 will satisfy the inequality.

### Conclusion

The positive integral values of \(x\) satisfying the inequality are \(2, 3, 4, \ldots\). Thus, there are infinitely many positive integral solutions.