Solve each inequality in Exercises 65–70 and graph the solution set on a real number line. 3/(x +3) > 3/(x - 2)

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Start by writing the inequality clearly: $\frac{3}{x + 3} > \frac{3}{x - 2}$.

Since both sides have a numerator of 3, you can focus on the denominators. However, be careful because multiplying or dividing by expressions involving variables can change the inequality direction depending on the sign of those expressions.

Bring all terms to one side to compare: $\frac{3}{x + 3} - \frac{3}{x - 2} > 0$. Find a common denominator to combine the fractions: $\frac{3(x - 2) - 3(x + 3)}{(x + 3)(x - 2)} > 0$.

Simplify the numerator: $3(x - 2) - 3(x + 3) = 3x - 6 - 3x - 9 = -15$. So the inequality becomes $\frac{-15}{(x + 3)(x - 2)} > 0$.

Analyze the sign of the fraction $\frac{-15}{(x + 3)(x - 2)}$. Since -15 is negative, the fraction is positive when the denominator is negative. Determine the intervals where $(x + 3)(x - 2) < 0$ by finding critical points at $x = -3$ and $x = 2$, then test intervals between and outside these points.

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