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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Solving Rational Inequalities

Rational inequalities involve expressions with variables in the denominator. To solve them, first bring all terms to one side to form a single rational expression, then determine where this expression is positive or negative by analyzing its critical points.

Finding Critical Points and Domain Restrictions

Critical points occur where the numerator or denominator equals zero. These points divide the number line into intervals to test. Also, values that make the denominator zero are excluded from the domain, as they cause undefined expressions.

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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