Solve each inequality. Give the solution set using interval notation. 3/x+2 > 2/x-4

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Start by rewriting the inequality clearly: $\frac{3}{x+2} > \frac{2}{x-4}$.

Bring all terms to one side to compare them: $\frac{3}{x+2} - \frac{2}{x-4} > 0$.

Find a common denominator, which is $(x+2)(x-4)$, and combine the fractions: $\frac{3(x-4) - 2(x+2)}{(x+2)(x-4)} > 0$.

Simplify the numerator: \$3(x-4) - 2(x+2) = 3x - 12 - 2x - 4 = (3x - 2x) + (-12 - 4) = x - 16$.

Rewrite the inequality as $\frac{x - 16}{(x+2)(x-4)} > 0$ and analyze the sign of the expression by identifying critical points $x = -2$, $x = 4$, and $x = 16$. Then test intervals between these points to determine where the expression is positive.

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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 find values that make denominators zero (excluded from the solution), then determine where the inequality holds by testing intervals defined by these 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 solution set because they cause undefined expressions.

Interval Notation for Solution Sets

Interval notation expresses solution sets as ranges of numbers using parentheses and brackets. Parentheses indicate values not included (open intervals), while brackets include endpoints. This notation concisely represents all values satisfying the inequality.