X+7)^9(x+8)(-4)^6<0

To solve the inequality \((x+7)^9(x+8)(-4)^6 < 0\), let's first analyze the components of the expression. 1. **Identify the sign of each factor**: - \((-4)^6\) is a positive constant because any even power of a negative number is positive. - The term \((x+7)^9\) is a polynomial that has a degree of 9, which means it will be: - Positive for \(x > -7\) - Zero at \(x = -7\) - Negative for \(x < -7\) - The term \((x+8)\) is a linear polynomial that will be: - Positive for \(x > -8\) - Zero at \(x = -8\) - Negative for \(x < -8\) 2. **Find the critical points**: - The critical points of the expression are where each factor is zero: - From \((x+7)^9\), we have \(x = -7\) - From \((x+8)\), we have \(x = -8\) 3. **Test the intervals determined by the critical points**: - The critical points divide the number line into the following intervals: \((-\infty, -8)\), \((-8, -7)\), and \((-7, \infty)\). - We will test a point in each interval to determine the sign of the expression in that interval. - **Interval \( (-\infty, -8) \)** (choose \(x = -9\)): \[ (-9+7)^9(-9+8) = (-2)^9(-1) = -(-2)^9 < 0 \] (Negative) - **Interval \( (-8, -7) \)** (choose \(x = -7.5\)): \[ (-7.5+7)^9(-7.5+8) = (-0.5)^9(0.5) = -0.5^{10} < 0 \] (Negative) - **Interval \( (-7, \infty) \)** (choose \(x = 0\)): \[ (0+7)^9(0+8) = 7^9(8) > 0 \] (Positive) 4. **Conclusion**: - The expression \((x+7)^9(x+8)(-4)^6 < 0\) holds true in the intervals where the product is negative, which is the interval \((-\infty, -8)\) and \((-8, -7)\). Thus, the solution to the inequality is: \[ (-\infty, -8) \cup (-8, -7) \]