Solve each rational inequality. Give the solution set in interval notation. 1 /{x² - 4x + 3} ≤ 1 /{ 3 - x}

Verified step by step guidance

1. Start by rewriting the inequality clearly: \(\frac{1}{x^{2} - 4x + 3} \leq \frac{1}{3 - x}\).
2. Identify the domain restrictions by finding values of \(x\) that make any denominator zero. Solve \(x^{2} - 4x + 3 = 0\) and \$3 - x = 0$ to find these points.
3. Bring all terms to one side to form a single rational inequality: \(\frac{1}{x^{2} - 4x + 3} - \frac{1}{3 - x} \leq 0\).
4. Find a common denominator, which is the product of the two denominators, and combine the fractions into a single rational expression: \(\frac{(3 - x) - (x^{2} - 4x + 3)}{(x^{2} - 4x + 3)(3 - x)} \leq 0\).
5. Simplify the numerator and factor both numerator and denominator if possible. Then determine the critical points by setting numerator and denominator equal to zero. Use these points to test intervals on the number line to find where the inequality holds true, remembering to exclude points where the denominator is zero. Finally, express the solution set in interval notation.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions with variables in the denominator. Solving them requires finding values of the variable that make the inequality true, while ensuring denominators are not zero to avoid undefined expressions.

Domain Restrictions

When dealing with rational expressions, it is crucial to identify values that make denominators zero, as these are excluded from the solution set. Determining the domain helps avoid invalid solutions and ensures the inequality is properly defined.

Interval Notation and Sign Analysis

After finding critical points from numerators and denominators, the number line is divided into intervals. Testing each interval determines where the inequality holds true. The solution is then expressed in interval notation, clearly showing all valid values.