(x+8)(x-1)<0
To solve the inequality \((x+8)(x-1)<0\), we will follow these steps:
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: \[ x + 8 = 0 \implies x = -8 \] \[ x - 1 = 0 \implies x = 1 \]
Determine the intervals: The critical points divide the number line into intervals. The intervals are:
- \( (-\infty, -8) \)
- \( (-8, 1) \)
- \( (1, \infty) \)
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 \( x = 2 \): \[ (2 + 8)(2 - 1) = (10)(1) = 10 > 0 \]
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 \( (-8, 1) \).
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 \((x+8)(x-1)<0\).
