Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9. (x+2)/(x+7)≥0

Rational inequalities involve expressions that are ratios of polynomials set in relation to an inequality (e.g., ≥, ≤, >, <). To solve them, one must determine where the rational expression is positive or negative, which often requires finding critical points where the numerator or denominator equals zero.

Critical points are values of the variable that make the rational expression equal to zero or undefined. For the inequality (x+2)/(x+7)≥0, the critical points are found by setting the numerator (x+2) to zero and the denominator (x+7) to zero, leading to x = -2 and x = -7. These points divide the number line into intervals for testing.

Interval notation is a way of representing the solution set of inequalities using intervals. It uses brackets [ ] for inclusive endpoints and parentheses ( ) for exclusive endpoints. For example, the solution set for the inequality can be expressed as intervals that indicate where the rational expression is non-negative.