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

Verified step by step guidance

1. Identify the critical points by setting the numerator and denominator equal to zero separately: solve \(x + 2 = 0\) and \(x + 7 = 0\). These points divide the number line into intervals.
2. Determine the intervals based on the critical points found: \(x = -2\) and \(x = -7\). The intervals are \((-infty, -7)\), \((-7, -2)\), and \((-2, infty)\).
3. Test a sample value from each interval in the inequality \(\frac{x+2}{x+7} \geq 0\) to check whether the expression is positive or zero in that interval.
4. Consider the points where the numerator is zero (\(x = -2\)) because the expression equals zero there, which satisfies the inequality \(\geq 0\). Exclude points where the denominator is zero (\(x = -7\)) because the expression is undefined there.
5. Combine the intervals where the inequality holds true and express the solution set in interval notation, including points where the expression equals zero but excluding points where it is undefined.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares this ratio to zero or another value. Solving them requires finding where the expression is positive, negative, or zero, considering the domain restrictions where the denominator is not zero.

Critical Points and Sign Analysis

Critical points are values that make the numerator or denominator zero, dividing the number line into intervals. By testing values in each interval, you determine the sign of the rational expression, which helps identify where the inequality holds true.

Interval Notation and Domain Restrictions

Interval notation expresses solution sets compactly using parentheses and brackets to indicate open or closed intervals. When solving rational inequalities, it is essential to exclude points where the denominator is zero, as these are not in the domain and cannot be part of the solution.