Question

Solve each rational inequality. Give the solution set in interval notation. 3/(2x-1)\u003e-4/x

Accepted Answer

Rewrite the inequality to have zero on one side by subtracting the right-hand side from both sides: $$\frac{3}{2x-1} - \left(-\frac{4}{x}\right) \u003e 0$$, which simplifies to $$\frac{3}{2x-1} + \frac{4}{x} \u003e 0. $$Find a common denominator for the two fractions, which is $$x(2x-1)$$, and combine the fractions into a single rational expression: $$\frac{3x + 4(2x - 1)}{x(2x - 1)} \u003e 0. $$Simplify the numerator: $$3x + 8x - 4 = 11x - 4$$, so the inequality becomes $$\frac{11x - 4}{x(2x - 1)} \u003e 0. $$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 $$\frac{11x - 4}{x(2x - 1)} \u003e 0 to $$find where the expression is positive, and then express the solution set in interval notation, excluding points where the denominator is zero.

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

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Key Concepts

Rational Inequalities

Finding a Common Denominator and Combining Terms

Sign Analysis and Interval Testing