Solve each rational inequality. Give the solution set in interval notation. 1 /(x+ 2) > 1 /(x -3)

Verified step by step guidance

1. Start by writing the inequality clearly: \(\frac{1}{x + 2} > \frac{1}{x - 3}\).
2. Bring all terms to one side to compare to zero: \(\frac{1}{x + 2} - \frac{1}{x - 3} > 0\).
3. Find a common denominator and combine the fractions: \(\frac{(x - 3) - (x + 2)}{(x + 2)(x - 3)} > 0\).
4. Simplify the numerator: \(\frac{x - 3 - x - 2}{(x + 2)(x - 3)} = \frac{-5}{(x + 2)(x - 3)} > 0\).
5. Analyze the inequality \(\frac{-5}{(x + 2)(x - 3)} > 0\) by considering the sign of the denominator and the fact that the numerator is a constant negative number; determine intervals where the entire expression is positive, and exclude values that make the denominator zero (i.e., \(x \neq -2\) and \(x \neq 3\)).

Key Concepts

Rational Inequalities

Rational inequalities involve expressions with variables in the denominator. Solving them requires finding values of the variable that make the inequality true, while ensuring the denominator is never zero to avoid undefined expressions.

Finding a Common Denominator and Combining Fractions

To compare or combine rational expressions, rewrite them with a common denominator. This allows you to subtract or add the fractions and transform the inequality into a single rational expression, simplifying the problem.

Sign Analysis and Interval Testing

After simplifying the inequality, determine where the expression is positive or negative by identifying critical points (zeros and undefined points). Test intervals between these points to find where the inequality holds, then express the solution in interval notation.