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

Verified step by step guidance

1. Start by writing the inequality clearly: \(\frac{1}{x - 1} < \frac{1}{x + 1}\).
2. Bring all terms to one side to compare them: \(\frac{1}{x - 1} - \frac{1}{x + 1} < 0\).
3. Find a common denominator and combine the fractions: \(\frac{(x + 1) - (x - 1)}{(x - 1)(x + 1)} < 0\).
4. Simplify the numerator: \(\frac{x + 1 - x + 1}{(x - 1)(x + 1)} = \frac{2}{(x - 1)(x + 1)} < 0\).
5. Analyze the sign of the expression \(\frac{2}{(x - 1)(x + 1)}\) by considering the critical points \(x = 1\) and \(x = -1\), and determine where the expression is negative. Remember to exclude values that make the denominator zero.

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 solve inequalities with rational expressions, rewrite both sides with a common denominator. This allows combining the inequality into a single rational expression, making it easier to analyze the sign of the numerator and denominator.

Sign Analysis and Interval Notation

After simplifying, determine where the rational expression is positive or negative by analyzing critical points (zeros of numerator and denominator). Use this to identify solution intervals, and express the solution set clearly using interval notation.