Solve the following inequality. \[ (x+4)(x+9)(5-x)<0 \] Write your answer as an interval or union of intervals. If there is no real solution, click on "No solution".

To solve the inequality \((x+4)(x+9)(5-x) < 0\), we first find the points where the expression equals zero: 1. \(x + 4 = 0 \implies x = -4\) 2. \(x + 9 = 0 \implies x = -9\) 3. \(5 - x = 0 \implies x = 5\) This gives us three critical points: \(-9\), \(-4\), and \(5\). We will use these points to test the sign of the expression in the intervals they divide the number line into: 1. **Interval** \((-∞, -9)\): Choose \(x = -10\) \(((-10 + 4)((-10 + 9)(5 - (-10)) = (-6)(-1)(15) > 0\) 2. **Interval** \((-9, -4)\): Choose \(x = -6\) \(((-6 + 4)((-6 + 9)(5 - (-6)) = (-2)(3)(11) < 0\) 3. **Interval** \((-4, 5)\): Choose \(x = 0\) \(((0 + 4)((0 + 9)(5 - 0)) = (4)(9)(5) > 0\) 4. **Interval** \((5, ∞)\): Choose \(x = 6\) \(((6 + 4)((6 + 9)(5 - 6)) = (10)(15)(-1) < 0\) Now, let's compile the intervals where the product is negative: - The product is negative in \((-9, -4)\) and \((5, ∞)\). Thus, the solution to the inequality \((x+4)(x+9)(5-x)<0\) is: \[ (-9, -4) \cup (5, \infty) \]