Answer:
To find the solution of the inequality (x + 8) / (5x - 1) > 0, we can follow these steps:
Find the critical points of the inequality by setting the numerator and denominator equal to zero. In this case, we have: x + 8 = 0 (critical point 1) 5x - 1 = 0 (critical point 2)
Solving these equations, we get: x = -8 (critical point 1) x = 1/5 (critical point 2)
Create a sign chart using the critical points. We divide the number line into three regions based on the critical points (-∞, -8), (-8, 1/5), and (1/5, +∞).
Region 1: Test a value less than -8, e.g., x = -10: Substitute x = -10 into the inequality: (-10 + 8) / (5(-10) - 1) > 0 Simplifying, we get: -2 / (-51) > 0 The result is positive (+), so the inequality is satisfied in Region 1.
Region 2: Test a value between -8 and 1/5, e.g., x = 0: Substitute x = 0 into the inequality: (0 + 8) / (5(0) - 1) > 0 Simplifying, we get: 8 / (-1) < 0 The result is negative (-), so the inequality is not satisfied in Region 2.
Region 3: Test a value greater than 1/5, e.g., x = 1: Substitute x = 1 into the inequality: (1 + 8) / (5(1) - 1) > 0 Simplifying, we get: 9 / 4 > 0 The result is positive (+), so the inequality is satisfied in Region 3.
Analyze the sign chart to determine the solution:
The inequality is satisfied in Region 1 (-∞, -8) and Region 3 (1/5, +∞) where the expression is greater than zero.
The inequality is not satisfied in Region 2 (-8, 1/5) where the expression is less than zero.
Therefore, the solution to the inequality (x + 8) / (5x - 1) > 0 is: x < -8 or x > 1/5