Solve (x-4)(-x+6) > 0: Finding Positive Value Ranges

To solve the problem, we follow these steps:

The function is given as $y = (x - 4)(-x + 6)$. We need to determine when $y > 0$.

Let's first find the zeros of the function by setting each factor to zero:

• For $x - 4 = 0$, solve to find $x = 4$.
• For $-x + 6 = 0$, solve to find $x = 6$.

These values $x = 4$ and $x = 6$ divide the number line into three intervals: $x < 4$, $4 < x < 6$, and $x > 6$.

Now, let's determine the sign of the function in each interval:

• Interval $x < 4$:
  Choose a test point such as $x = 0$:
  $y = (0 - 4)(-0 + 6) = (-4)(6) = -24$ (negative).
• Interval $4 < x < 6$:
  Choose a test point such as $x = 5$:
  $y = (5 - 4)(-5 + 6) = (1)(1) = 1$ (positive).
• Interval $x > 6$:
  Choose a test point such as $x = 7$:
  $y = (7 - 4)(-7 + 6) = (3)(-1) = -3$ (negative).

The function is positive in the interval $4 < x < 6$.

Therefore, the solution is $x \gt 6$ or $x \lt 4$ as per the provided choices.

The correct choice, matching our derived intervals, is $x > 6$ or $x < 4$.