The number of integral value(s) of x satisfying the equation ∣ ∣ ∣ ​ x ^ (4) ⋅3 ∣x−2∣ ⋅5 ^(x−1) ∣ ∣ ∣ ​ =−x ^(4) ⋅3 ^(∣x−2∣) ⋅5 ^(x−1) , is

Options

A. a. 2
B. b. 3
C. c. 1
D. d. infinite

Answer

To solve the equation

\[ \left| x^4 \cdot 3^{|x-2|} \cdot 5^{(x-1)} \right| = -x^4 \cdot 3^{|x-2|} \cdot 5^{(x-1)}, \]

we will analyze both sides step by step.

### Step 1: Analyze the left-hand side
The left-hand side of the equation is

\[ \left| x^4 \cdot 3^{|x-2|} \cdot 5^{(x-1)} \right|. \]

Since the absolute value of any expression is always non-negative, we have:

\[ \left| x^4 \cdot 3^{|x-2|} \cdot 5^{(x-1)} \right| \geq 0. \]

### Step 2: Analyze the right-hand side
The right-hand side of the equation is

\[ -x^4 \cdot 3^{|x-2|} \cdot 5^{(x-1)}. \]

This expression can be negative or zero depending on the value of \(x\).

### Step 3: Set conditions for equality
For the equation to hold, we need:

\[ \left| x^4 \cdot 3^{|x-2|} \cdot 5^{(x-1)} \right| = -x^4 \cdot 3^{|x-2|} \cdot 5^{(x-1)}. \]

Since the left-hand side is always non-negative, the right-hand side must also be non-positive. Therefore, both sides can only be equal if they are equal to zero:

\[ \left| x^4 \cdot 3^{|x-2|} \cdot 5^{(x-1)} \right| = 0. \]

### Step 4: Solve for when the left-hand side equals zero
The left-hand side equals zero when:

\[ x^4 \cdot 3^{|x-2|} \cdot 5^{(x-1)} = 0. \]

Since \(3^{|x-2|}\) and \(5^{(x-1)}\) are both positive for all real \(x\), the only way for the product to equal zero is if:

\[ x^4 = 0. \]

### Step 5: Find integral solutions
The equation \(x^4 = 0\) has only one solution:

\[ x = 0. \]

### Conclusion
Thus, the number of integral values of \(x\) satisfying the original equation is:

\[ \boxed{1}. \]