Final answer:
To solve the inequality (5x+8/(x^2+1))<0, find the critical points and determine the intervals where the expression is positive or negative. The solution in interval notation is (-∞, -8/5) U (-8/5, ∞).
Step-by-step explanation:
To solve the inequality (5x+8/(x^2+1))<0, we need to find the critical points of the expression and determine the intervals where it is positive or negative.
First, let's find the critical points by setting the numerator equal to zero: 5x + 8 = 0. Solving for x, we get x = -8/5. Note that the denominator x^2 + 1 is always positive.
Next, we construct a sign chart, testing the sign of the expression for test values in each interval. We choose test values -10, 0, and 10. Evaluating the expression for these values, we find that it is positive in the intervals (-∞, -8/5) and (-8/5, ∞). Therefore, the solution to the inequality in interval notation is (-∞, -8/5) U (-8/5, ∞).