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

Verified step by step guidance

1. Start by writing the inequality clearly: \(\frac{5}{2 - x} > \frac{3}{3 - x}\).
2. Find a common denominator to combine the fractions or cross-multiply, but be careful about the signs since the denominators can be positive or negative. The denominators are \$2 - x\( and \)3 - x\(, so note where these expressions are zero to determine domain restrictions: \)x \neq 2\( and \)x \neq 3$.
3. Rewrite the inequality by bringing all terms to one side: \(\frac{5}{2 - x} - \frac{3}{3 - x} > 0\).
4. Combine the two fractions over a common denominator: \(\frac{5(3 - x) - 3(2 - x)}{(2 - x)(3 - x)} > 0\).
5. Simplify the numerator and analyze the sign of the entire expression by considering the critical points from the numerator and denominator. Then, create a sign chart to determine where the expression is positive, keeping in mind the domain restrictions \(x \neq 2\) and \(x \neq 3\).

Key Concepts

Rational Inequalities

Rational inequalities involve expressions with variables in the denominator. To solve them, you must find values of the variable that make the inequality true while ensuring the denominator is not zero, as division by zero is undefined.

Finding a Common Denominator and Combining Terms

To compare or combine rational expressions, rewrite them with a common denominator. This allows you to create a single inequality involving a rational expression, which can then be analyzed by examining the numerator and denominator separately.

Sign Analysis and Interval Notation

After simplifying, determine where the rational expression is positive or negative by testing intervals defined by critical points (zeros of numerator and denominator). The solution set is expressed in interval notation, indicating all values satisfying the inequality.