Solve each inequality. Give the solution set using interval notation. 3/x+2 > 2/x-4

Verified step by step guidance

1. Start by writing the inequality clearly: \(\frac{3}{x+2} > \frac{2}{x-4}\).
2. Find a common denominator to combine the fractions or cross-multiply, but be careful about the domain restrictions where denominators are zero: \(x \neq -2\) and \(x \neq 4\).
3. Rewrite the inequality by bringing all terms to one side: \(\frac{3}{x+2} - \frac{2}{x-4} > 0\).
4. Combine the fractions over a common denominator: \(\frac{3(x-4) - 2(x+2)}{(x+2)(x-4)} > 0\).
5. Simplify the numerator and analyze the sign of the rational expression by considering critical points from numerator and denominator to determine intervals where the inequality holds, then express the solution set in interval notation.

Key Concepts

Solving Rational Inequalities

Rational inequalities involve expressions with variables in the denominator. To solve them, first find a common denominator, rewrite the inequality as a single rational expression, and determine where the expression is positive or negative by analyzing critical points.

Critical Points and Sign Analysis

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

Interval Notation

Interval notation is a concise way to represent solution sets on the number line. It uses parentheses for values not included (open intervals) and brackets for included values (closed intervals), clearly showing the range of solutions.