Graph the solution to the following inequality on the number line. \[ (x-4)(x+5)<0 \]

To solve the inequality \((x-4)(x+5)<0\), we first need to find the points where the expression equals zero. This occurs when either factor is zero: 1. \(x - 4 = 0 \Rightarrow x = 4\) 2. \(x + 5 = 0 \Rightarrow x = -5\) These points, \(x = -5\) and \(x = 4\), divide the number line into three intervals: 1. \( (-\infty, -5) \) 2. \( (-5, 4) \) 3. \( (4, \infty) \) Next, we will test a point from each interval to determine where the product \((x-4)(x+5)\) is negative. 1. **Interval \( (-\infty, -5) \)**: Choose \(x = -6\) \[ (-6 - 4)(-6 + 5) = (-10)(-1) = 10 \quad (\text{positive}) \] 2. **Interval \( (-5, 4) \)**: Choose \(x = 0\) \[ (0 - 4)(0 + 5) = (-4)(5) = -20 \quad (\text{negative}) \] 3. **Interval \( (4, \infty) \)**: Choose \(x = 5\) \[ (5 - 4)(5 + 5) = (1)(10) = 10 \quad (\text{positive}) \] From our tests, we find that the product \((x-4)(x+5)\) is negative in the interval \((-5, 4)\). Now, we need to express the solution to the inequality \((x-4)(x+5)<0\). Since the inequality is strict (less than, not less than or equal to), we do not include the endpoints \(-5\) and \(4\). Thus, the solution is: \[ (-5, 4) \] ### Graphing the Solution on the Number Line To graph this solution on a number line: 1. Draw a number line. 2. Mark the points \(-5\) and \(4\). 3. Use open circles at \(-5\) and \(4\) to indicate that these points are not included in the solution. 4. Shade the region between \(-5\) and \(4\) to indicate that all numbers in this interval are part of the solution. The final graph will look like this: ``` <---o=================o---> -5 4 ``` Where the shaded area represents the solution \((-5, 4)\).