Solve without a calculator. Show your factoring and a complete number-line on your worksheet along with the required information and shading or rough graph with shading. Write your solution using interval notation. \( x^{3}<8 x^{2} \) Solution in interval notation:

To solve the inequality \( x^{3} < 8x^{2} \), let's start by rearranging it into standard form: 1. Bring all terms to one side: \[ x^{3} - 8x^{2} < 0 \] 2. Factor out the common term: \[ x^{2}(x - 8) < 0 \] Now we can find the critical points by setting \( x^{2} = 0 \) or \( x - 8 = 0 \): - From \( x^{2} = 0 \): \( x = 0 \) - From \( x - 8 = 0 \): \( x = 8 \) Critical points are at \( x = 0 \) and \( x = 8 \). These points divide the number line into three intervals: \( (-\infty, 0) \), \( (0, 8) \), and \( (8, \infty) \). Now, we’ll test each interval to see where the inequality holds true: 1. **Interval \( (-\infty, 0) \)**: Choose \( x = -1 \) \[ (-1)^{2}(-1 - 8) = 1 \cdot (-9) < 0 \quad \text{(True)} \] 2. **Interval \( (0, 8) \)**: Choose \( x = 1 \) \[ (1)^{2}(1 - 8) = 1 \cdot (-7) < 0 \quad \text{(True)} \] 3. **Interval \( (8, \infty) \)**: Choose \( x = 9 \) \[ (9)^{2}(9 - 8) = 81 \cdot 1 > 0 \quad \text{(False)} \] The inequality holds for the intervals \( (-\infty, 0) \) and \( (0, 8) \). Now, let’s put this all together in interval notation: The solution is: \[ (-\infty, 0) \cup (0, 8) \] On the number line, you'd shade the regions corresponding to these intervals and leave points at \( x = 0 \) and \( x = 8 \) unshaded since these points do not satisfy the strict inequality.