Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation.1/(x - 3) < 1

Verified step by step guidance

1. Start by rewriting the inequality: \(\frac{1}{x - 3} < 1\).
2. Bring all terms to one side to have zero on the other side: \(\frac{1}{x - 3} - 1 < 0\).
3. Find a common denominator and combine the terms: \(\frac{1 - (x - 3)}{x - 3} < 0\).
4. Simplify the numerator: \(\frac{1 - x + 3}{x - 3} < 0\), which becomes \(\frac{4 - x}{x - 3} < 0\).
5. Determine the critical points by setting numerator and denominator equal to zero: numerator \$4 - x = 0\( gives \)x = 4\(, denominator \)x - 3 = 0\( gives \)x = 3$. Use these points to test intervals on the number line and find where the inequality holds true.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one or both sides contain rational functions, which are ratios of polynomials. Solving them requires finding values of the variable that make the inequality true, often by analyzing the sign of the expression over different intervals.

Critical Points and Sign Analysis

Critical points are values where the rational expression is zero or undefined, such as where the numerator or denominator equals zero. These points divide the number line into intervals, and testing each interval helps determine where the inequality holds true.

Interval Notation and Graphing Solutions

Interval notation is a concise way to express solution sets using parentheses and brackets to indicate open or closed intervals. Graphing on a number line visually represents these intervals, showing where the inequality is satisfied and highlighting excluded points.