Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9. 1/(x+2)≥3

Verified step by step guidance

1. Start by rewriting the inequality: \(\frac{1}{x+2} \geq 3\).
2. Bring all terms to one side to have zero on the other side: \(\frac{1}{x+2} - 3 \geq 0\).
3. Find a common denominator and combine the terms into a single rational expression: \(\frac{1 - 3(x+2)}{x+2} \geq 0\).
4. Simplify the numerator: \(\frac{1 - 3x - 6}{x+2} = \frac{-3x - 5}{x+2} \geq 0\).
5. Determine the critical points by setting numerator and denominator equal to zero separately: solve \(-3x - 5 = 0\) and \(x + 2 = 0\), then analyze the sign of the expression on intervals defined by these points to find where the inequality holds.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where variables appear in the denominator. Solving them requires finding values of the variable that make the inequality true, while ensuring the denominator is not zero to avoid undefined expressions.

Critical Points and Sign Analysis

Critical points are values where the numerator or denominator equals zero, dividing the number line into intervals. Testing each interval helps determine where the inequality holds by checking the sign of the expression in those intervals.

Interval Notation

Interval notation is a concise way to represent solution sets of inequalities. It uses parentheses for excluded endpoints and brackets for included endpoints, clearly showing the range of values satisfying the inequality.