Solve each rational inequality. Give the solution set in interval notation.
$\frac{5}{2-x}>\frac{3}{3-x}$

Verified step by step guidance

1. Start with the given inequality: \(\frac{5}{2 - x} > \frac{3}{3 - x}\).
2. Identify the domain restrictions by setting the denominators not equal to zero: \(2 - x \neq 0\) and \(3 - x \neq 0\), which means \(x \neq 2\) and \(x \neq 3\).
3. Bring all terms to one side to form a single rational expression: \(\frac{5}{2 - x} - \frac{3}{3 - x} > 0\).
4. Find a common denominator, which is \((2 - x)(3 - x)\), and combine the fractions: \(\frac{5(3 - x) - 3(2 - x)}{(2 - x)(3 - x)} > 0\).
5. Simplify the numerator and analyze the sign of the rational expression by considering the critical points from the numerator and denominator to determine the solution intervals.

Key Concepts

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 ensuring the denominator is never zero to avoid undefined expressions.

Finding a Common Denominator and Cross-Multiplication

To compare two rational expressions, you can cross-multiply when denominators are positive or find a common denominator to combine terms. Care must be taken to consider the sign of denominators, as multiplying or dividing by negative values reverses inequality signs.

Domain Restrictions and Interval Notation

The domain excludes values that make any denominator zero. After solving the inequality, solutions must be expressed in interval notation, clearly indicating which values satisfy the inequality and respecting domain restrictions.