Question

If \( |x - 6| = |x^2 - 4x| - |x^2 - 5x + 6| \), \(\text{ where \( x \) is a real variable.}\)

• A. \( x = (2, 5) \)
• B. \( x \in [2, 3] \cup [6, \infty) \)
• C. \( x \in \mathbb{R} - [2, 6] \)
• D. None of these

Hint:

For equations involving absolute values, break the equation into intervals based on the critical points where the expressions inside the absolute values change sign.

Answer 1

The Correct Option is B

Step 1: The given equation is: \[ |x - 6| = |x^2 - 4x| - |x^2 - 5x + 6| \] Step 2: Factor the quadratic expressions inside the absolute values: - \( x^2 - 4x = x(x - 4) \) - \( x^2 - 5x + 6 = (x - 2)(x - 3) \) Step 3: Analyze the intervals where each absolute value expression behaves differently. The critical points where the expressions inside absolute values change sign are \( x = 6, 4, 3, 2 \). Step 4: Check the behavior of the equation for different intervals of \( x \): - For \( x \in [2, 3] \cup [6, \infty) \), the equation holds true after simplification. Thus, the solution is \( x \in [2, 3] \cup [6, \infty) \), which corresponds to option (b).