Solve each rational inequality. Give the solution set in interval notation. 3 /{4 - x} > 6 /{ 1 - x}

Verified step by step guidance

1. Start by writing the inequality clearly: \(\frac{3}{4 - x} > \frac{6}{1 - x}\).
2. Identify the critical points where the denominators are zero, since these values are not in the domain. Solve \$4 - x = 0\( and \)1 - x = 0\( to find \)x = 4\( and \)x = 1$ respectively.
3. Bring all terms to one side to form a single rational expression: \(\frac{3}{4 - x} - \frac{6}{1 - x} > 0\).
4. Find a common denominator, which is \((4 - x)(1 - x)\), and combine the fractions: \(\frac{3(1 - x) - 6(4 - x)}{(4 - x)(1 - x)} > 0\).
5. Simplify the numerator and analyze the sign of the rational expression by considering the critical points and the zeros of the numerator. Use a sign chart to determine where the expression is positive, then express the solution set in interval notation excluding points where the denominator is zero.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions with variables in the denominator. Solving them requires finding values that make the inequality true while ensuring denominators are not zero. The solution often involves analyzing the sign of the rational expression over different intervals.

Critical Points and Sign Analysis

Critical points are values where the numerator or denominator equals zero, dividing the number line into intervals. By testing points in each interval, you determine where the inequality holds. This method helps identify solution sets for rational inequalities.

Interval Notation

Interval notation expresses solution sets as intervals on the number line, using parentheses for excluded endpoints and brackets for included ones. It provides a concise way to represent all values satisfying the inequality, especially when solutions are unions of multiple intervals.