Solve each inequality. Give the solution set using interval notation. 3x+6 / x-5 > 0

Verified step by step guidance

1. Identify the inequality to solve: \(\frac{3x + 6}{x - 5} > 0\).
2. Find the critical points by setting the numerator and denominator equal to zero separately: solve \$3x + 6 = 0\( and \)x - 5 = 0$.
3. Determine the critical points: \(x = -2\) from the numerator and \(x = 5\) from the denominator. These points divide the number line into intervals.
4. Test each interval determined by the critical points \((-\infty, -2)\), \((-2, 5)\), and \((5, \infty)\) by choosing a test value from each interval and substituting it into the inequality \(\frac{3x + 6}{x - 5} > 0\) to check if the expression is positive.
5. Based on the sign of the expression in each interval, write the solution set in interval notation, remembering to exclude \(x = 5\) because it makes the denominator zero and the expression undefined.

Key Concepts

Solving Rational Inequalities

Rational inequalities involve expressions with variables in the numerator and denominator. To solve them, identify where the expression is positive or negative by finding critical points where the numerator or denominator equals zero, then test intervals between these points.

Critical Points and Sign Analysis

Critical points occur where the numerator or denominator is zero, dividing the number line into intervals. By testing values from each interval in the inequality, you determine where the expression satisfies the inequality, considering domain restrictions from the denominator.

Interval Notation

Interval notation expresses solution sets as intervals on the number line, using parentheses for excluded endpoints and brackets for included ones. It concisely represents all values that satisfy the inequality, excluding points where the expression is undefined.