$x \times (|a| \times |b|) = -ab$

Concept & Logic This algebraic question deals with the properties of absolute values. The absolute value of a product is equal to the product of their absolute values: $|a| \times |b| = |ab|$. Step-by-Step Solution * Simplify the given equation: $$x \times (|a| \times |b|) = -ab$$ $$x \times |ab| = -ab$$ * Isolate $x$: $$x = \frac{-ab}{|ab|}$$ * The value of $x$ depends entirely on the sign of the product $ab$: * Case 1: If $a$ and $b$ have the SAME sign (both positive or both negative), then $ab > 0$. Therefore, $|ab| = ab$. $$x = \frac{-ab}{ab} = -1$$ * Case 2: If $a$ and $b$ have OPPOSITE signs (one positive, one negative), then $ab < 0$. Therefore, $|ab| = -ab$. $$x = \frac{-ab}{-ab} = 1$$ * Since the value of $x$ is not a single constant and changes based on the signs of $a$ and $b$, neither $1$ nor $-1$ is universally correct. Exam Strategy & Shortcut Whenever dealing with absolute values equal to variables without specified domains (e.g., $a,b > 0$), expect multiple cases. Since $x$ can evaluate to either $1$ or $-1$ depending on the integers chosen, a single definitive numeric option is impossible. Instantly choose "None of these". Common Pitfall Assuming that variables $a$ and $b$ are strictly positive integers. If you make this assumption, you will erroneously conclude that $x = -1$ and select option (b). Final Answer **Therefore, the correct answer is None of these.**