Solve each inequality. Give the solution set using interval notation. $\frac{3x-2}{x}-4>0$

Verified step by step guidance

1. Rewrite the inequality to have a single rational expression on one side: \(\frac{3x - 2}{x} - 4 > 0\).
2. Combine the terms into a single fraction by expressing 4 as \(\frac{4x}{x}\), so the inequality becomes \(\frac{3x - 2}{x} - \frac{4x}{x} > 0\).
3. Subtract the fractions to get a single rational expression: \(\frac{3x - 2 - 4x}{x} > 0\), which simplifies to \(\frac{-x - 2}{x} > 0\).
4. Identify the critical points by setting the numerator and denominator equal to zero: numerator \(-x - 2 = 0\) gives \(x = -2\), denominator \(x = 0\).
5. Use these critical points to divide the number line into intervals, then test a value from each interval in the inequality \(\frac{-x - 2}{x} > 0\) to determine where the inequality holds true. Finally, express the solution set in interval notation.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares this ratio to zero or another value. Solving them requires finding where the numerator and denominator change sign, considering restrictions where the denominator is zero.

Critical Points and Sign Analysis

Critical points are values that make the numerator or denominator zero, dividing the number line into intervals. By testing points in each interval, you determine the sign of the rational expression to identify where the inequality holds true.

Interval Notation

Interval notation is a concise way to express solution sets of inequalities, using parentheses for excluded endpoints and brackets for included ones. It clearly shows ranges of values that satisfy the inequality, especially important when dealing with domain restrictions.