(x+8)(x-1)<0

To solve the inequality \((x+8)(x-1)<0\), we will follow these steps: 1. **Find the critical points**: Set the inequality equal to zero and solve for \(x\). \[ (x+8)(x-1) = 0 \] This gives us the critical points: \[ x + 8 = 0 \implies x = -8 \] \[ x - 1 = 0 \implies x = 1 \] 2. **Determine the intervals**: The critical points divide the number line into intervals. The intervals are: - \( (-\infty, -8) \) - \( (-8, 1) \) - \( (1, \infty) \) 3. **Test each interval**: We can choose a test point from each interval to determine the sign of the product \((x + 8)(x - 1)\). - For the interval \( (-\infty, -8) \), let's test \( x = -9 \): \[ (-9 + 8)(-9 - 1) = (-1)(-10) = 10 > 0 \] - For the interval \( (-8, 1) \), let's test \( x = 0 \): \[ (0 + 8)(0 - 1) = (8)(-1) = -8 < 0 \] - For the interval \( (1, \infty) \), let's test \( x = 2 \): \[ (2 + 8)(2 - 1) = (10)(1) = 10 > 0 \] 4. **Compile the results**: From our tests, we have: - The product is positive in the intervals \( (-\infty, -8) \) and \( (1, \infty) \). - The product is negative in the interval \( (-8, 1) \). 5. **Write the solution**: Since we want the values where \((x + 8)(x - 1) < 0\), we conclude: \[ \boxed{(-8, 1)} \] This represents the set of \(x\) values that satisfy the original inequality \((x+8)(x-1)<0\).