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

Key Concepts

Inequalities

Inequalities are mathematical statements that express the relationship between two expressions that are not equal. They can be represented using symbols such as '>', '<', '≥', and '≤'. Solving inequalities involves finding the values of the variable that make the inequality true, which often requires manipulating the expressions similarly to equations but with special attention to the direction of the inequality when multiplying or dividing by negative numbers.

Interval Notation

Interval notation is a way of representing a set of numbers between two endpoints. It uses parentheses and brackets to indicate whether the endpoints are included in the set. For example, (a, b) means all numbers between a and b, excluding a and b, while [a, b] includes both endpoints. This notation is particularly useful for expressing solution sets of inequalities succinctly.

Rational Expressions

Rational expressions are fractions where the numerator and denominator are polynomials. When solving inequalities involving rational expressions, it is crucial to identify the values that make the denominator zero, as these points create restrictions in the solution set. Additionally, understanding how to manipulate and simplify these expressions is essential for finding the critical points that help determine the intervals to test for the inequality.