Solve each inequality in Exercises 65–70 and graph the solution set on a real number line. 1/(x + 1) > 2/(x - 1)

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Start by writing the inequality clearly: $\frac{1}{x + 1} > \frac{2}{x - 1}$.

Bring all terms to one side to have zero on the other side: $\frac{1}{x + 1} - \frac{2}{x - 1} > 0$.

Find a common denominator, which is $(x + 1)(x - 1)$, and combine the fractions: $\frac{(x - 1) - 2(x + 1)}{(x + 1)(x - 1)} > 0$.

Simplify the numerator: $(x - 1) - 2(x + 1) = x - 1 - 2x - 2 = -x - 3$, so the inequality becomes $\frac{-x - 3}{(x + 1)(x - 1)} > 0$.

Determine the critical points by setting numerator and denominator equal to zero: numerator $-x - 3 = 0$ gives $x = -3$, denominator $(x + 1)(x - 1) = 0$ gives $x = -1$ and $x = 1$. Use these points to test intervals on the number line to find where the inequality holds.

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Graphing Solution Sets on a Number Line

After determining intervals where the inequality holds, represent the solution set on a real number line. Use open or closed circles to indicate whether endpoints are included or excluded, and shade the regions that satisfy the inequality.

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