Final answer:
To solve the inequality y² + 7y - 8 ≥ 0, we factor the quadratic, find its zeros, and test intervals to determine where the inequality holds true. The solution in interval notation is [−8, 1) ∪ (1, ∞).
Step-by-step explanation:
To solve the inequality y² + 7y - 8 ≥ 0, we first need to find the zeros of the quadratic by factoring it into (y+8)(y-1) = 0, which gives us two solutions: y = -8 and y = 1. These are the points where the graph of the quadratic equation touches the x-axis. To determine the intervals where the inequality is satisfied, we test points from each interval determined by the zeros.
First, we pick a value less than -8, for example y = -9, and substitute it into the inequality to check if it satisfies the condition:
(-9)² + 7(-9) - 8 = 81 - 63 - 8 = 10, which is greater than 0, so the interval (−∞, -8) satisfies the inequality.
Next, pick a value between -8 and 1, for example y = 0, and check it:
(0)² + 7(0) - 8 = - 8, which is not greater than 0, so the interval (−8, 1) does not satisfy the inequality.
Lastly, we choose a value greater than 1, for example y = 2, and substitute it into the inequality to check if it satisfies the condition:
(2)² + 7(2) - 8 = 4 + 14 - 8 = 10, which is greater than 0, so the interval (1, ∞) satisfies the inequality.
Combining the two satisfying intervals, the solution in interval notation is [−8, 1) ∪ (1, ∞).