Answer:
x ∈ (-∞, -3) ∪ (1, ∞)
Explanation:
To solve the inequality (1/(x^2+2x-3) < 0), we can start by finding the roots of the quadratic expression in the denominator, x^2 + 2x - 3, and determining the intervals where the expression is positive or negative.
The quadratic can be factored as (x + 3)(x - 1), so the roots are x = -3 and x = 1.
Now, we can create a sign chart to determine the intervals where the expression is positive or negative:
-3 1
|-------|-------|
- + +
In the interval (-∞, -3), the expression (x^2+2x-3) is negative. In the interval (-3, 1), the expression is positive, and in the interval (1, ∞), it is negative again.
Since we are looking for the inequality (1/(x^2+2x-3) < 0), we are interested in the intervals where the expression is negative.
Based on the sign chart, the solution to the inequality is (-∞, -3) ∪ (1, ∞).
Therefore, the quadratic is alone on a single inequality as:
x ∈ (-∞, -3) ∪ (1, ∞)