Solve each rational inequality. Give the solution set in interval notation. 3/(x+5)>2

Verified step by step guidance

1. Start by rewriting the inequality to have zero on one side: \(\frac{3}{x+5} - 2 > 0\).
2. Find a common denominator to combine the terms: \(\frac{3}{x+5} - \frac{2(x+5)}{x+5} > 0\).
3. Simplify the numerator: \(\frac{3 - 2(x+5)}{x+5} > 0\), which becomes \(\frac{3 - 2x - 10}{x+5} > 0\) or \(\frac{-2x - 7}{x+5} > 0\).
4. Identify critical points by setting numerator and denominator equal to zero: numerator \(-2x - 7 = 0\) and denominator \(x + 5 = 0\). Solve these to find \(x = -\frac{7}{2}\) and \(x = -5\).
5. Use these critical points to divide the number line into intervals, then test a value from each interval in the inequality \(\frac{-2x - 7}{x+5} > 0\) to determine where the inequality holds true. Remember to exclude \(x = -5\) since it makes the denominator zero.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one side is a ratio of polynomials. Solving them requires finding values of the variable that make the inequality true, considering where the expression is defined and where it changes sign.

Domain Restrictions

Since rational expressions have denominators, values that make the denominator zero are excluded from the solution set. Identifying these restrictions is crucial to avoid undefined expressions and correctly determine the solution intervals.

Interval Testing and Sign Analysis

After finding critical points from the numerator and denominator, the number line is divided into intervals. Testing values from each interval helps determine where the inequality holds true, allowing the solution set to be expressed in interval notation.