Solve each rational inequality. Give the solution set in interval notation. 3/(2x-1)>-4/x

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Rewrite the inequality to have zero on one side by subtracting the right-hand side from both sides: \(\frac{3}{2x-1} - \left(-\frac{4}{x}\right) > 0\), which simplifies to \(\frac{3}{2x-1} + \frac{4}{x} > 0\).

Find a common denominator for the two fractions, which is \(x(2x-1)\), and combine the fractions into a single rational expression: \(\frac{3x + 4(2x - 1)}{x(2x - 1)} > 0\).

Simplify the numerator: \(3x + 8x - 4 = 11x - 4\), so the inequality becomes \(\frac{11x - 4}{x(2x - 1)} > 0\).

Determine the critical points by setting the numerator and denominator equal to zero: numerator zero at \(11x - 4 = 0\), denominator zero at \(x = 0\) and \(2x - 1 = 0\). These points divide the number line into intervals to test.

Test each interval determined by the critical points in the inequality \(\frac{11x - 4}{x(2x - 1)} > 0\) to find where the expression is positive, and then express the solution set in interval notation, excluding points where the denominator is zero.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Rational Inequalities

Rational inequalities involve expressions where variables appear in the denominator. Solving them requires finding values of the variable that make the inequality true, while considering restrictions where the denominator is zero to avoid undefined expressions.

Finding a Common Denominator and Combining Terms

To solve rational inequalities, it is often necessary to rewrite both sides with a common denominator. This allows combining terms into a single rational expression, making it easier to analyze the inequality by examining the sign of the numerator and denominator.

Sign Analysis and Interval Testing

After combining terms, the solution involves determining where the rational expression is positive or negative. This is done by identifying critical points (zeros of numerator and denominator) and testing intervals between these points to see where the inequality holds.