Final answer:
To solve the inequality x²+3x-4≥0, factor the quadratic to (x+4)(x-1)≥0. The solution is the set of x values in the intervals x ≤ -4 or x ≥ 1, based on testing values around the critical points x = -4 and x = 1.
Step-by-step explanation:
To solve the inequality x²+3x-4≥0, we can factor the quadratic expression into its binomial factors. The factors of -4 that add up to 3 are 4 and -1. So, we can write the quadratic expression as (x+4)(x-1). The inequality becomes (x+4)(x-1)≥0. To solve this inequality, we need to find the values of x where the expression is zero or positive.
Setting each factor equal to zero gives us the critical values x = -4 and x = 1. The quadratic is zero at these values of x and the inequality holds true. To find the intervals where the quadratic expression is positive, we need to test intervals around these critical values.
Testing an x-value less than -4, say x = -5, in the inequality we get (x+4)(x-1) > 0, which evaluates to true. This shows that the interval (-∞, -4) is part of the solution. Testing an x-value between -4 and 1, say x = 0, the result is false, indicating that the interval (-4, 1) is not in our solution set. Lastly, testing an x-value greater than 1, say x = 2, returns true, including the interval (1, ∞) in our solution set. So, the solution to the inequality x²+3x-4≥0 is x ≤ -4 or x ≥ 1.