Solve each rational inequality. Give the solution set in interval notation. See Examples 4 and 5. 5 /{2 - x} > 3 /{ 3 - x}

Verified step by step guidance

1. Start by finding a common denominator for the rational expressions on both sides of the inequality. The denominators are \(2 - x\) and \(3 - x\).
2. Multiply both sides of the inequality by the common denominator \((2 - x)(3 - x)\) to eliminate the fractions.
3. Simplify the resulting inequality. Be careful with the signs, especially since multiplying or dividing by a negative number will reverse the inequality sign.
4. Determine the critical points by setting each factor in the denominator equal to zero: \(2 - x = 0\) and \(3 - x = 0\). These points are \(x = 2\) and \(x = 3\).
5. Test intervals around the critical points to determine where the inequality holds true. Use these intervals to write the solution set in interval notation, considering any restrictions from the original inequality.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials set in an inequality format. To solve them, one must determine where the rational expression is greater than or less than zero. This often requires finding critical points where the expression is undefined or equals zero, and then testing intervals to see where the inequality holds true.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, the interval (2, 5] includes all numbers greater than 2 and up to 5, including 5 but not 2.

Critical Points

Critical points are values of the variable where the rational expression is either zero or undefined. These points are essential for determining the intervals to test when solving rational inequalities. By analyzing the sign of the expression around these critical points, one can establish where the inequality is satisfied.