Solve (x-4)(-x+6): Finding Values Where Function is Positive

To solve this problem, we'll first find the roots of the function $y = (x-4)(-x+6)$ to determine the intervals that we need to examine.

• Step 1: Find the roots. Set the function equal to zero: $(x-4)(-x+6) = 0$.
• For $x-4=0$, we get $x = 4$.
• For $-x+6=0$, we get $x = 6$.
• Step 2: Identify the intervals created by these roots: $x < 4$, $4 < x < 6$, and $x > 6$.
• Step 3: Test a point in each interval to determine the sign of the function.
• Select a point from $x < 4$, say $x = 0$: $(0-4)(-0+6) = (-4)(6) = -24$, which is negative.
• Select a point from $4 < x < 6$, say $x = 5$: $(5-4)(-5+6) = (1)(1) = 1$, which is positive.
• Select a point from $x > 6$, say $x = 7$: $(7-4)(-7+6) = (3)(-1) = -3$, which is negative.
• Step 4: Conclude that the function is positive in the interval $4 < x < 6$.

Therefore, the values of $x$ for which $f(x) > 0$ are those in the interval $\mathbf{4 < x < 6}$.