Solve each rational inequality. Give the solution set in interval notation. 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: 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 that satisfy the inequality while ensuring the denominator is not zero, as division by zero is undefined.

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 true by analyzing the sign of the expression.

Interval Notation

Interval notation expresses solution sets using brackets and parentheses to indicate inclusive or exclusive endpoints. It concisely represents all values satisfying the inequality, excluding points where the expression is undefined.