Solve each rational inequality. Give the solution set in interval notation. (x - 8)/(x - 4) < 3

Verified step by step guidance

1. Start by rewriting the inequality to have zero on one side: subtract 3 from both sides to get \(\frac{(x - 8)}{(x - 4)} - 3 < 0\).
2. Find a common denominator to combine the terms into a single rational expression: write \(3\) as \(\frac{3(x - 4)}{(x - 4)}\) and subtract to get \(\frac{(x - 8) - 3(x - 4)}{(x - 4)} < 0\).
3. Simplify the numerator by distributing and combining like terms: calculate \((x - 8) - 3(x - 4)\) to get a simplified numerator expression.
4. Determine the critical points by setting the numerator equal to zero and the denominator equal to zero separately. These points divide the number line into intervals to test.
5. Test each interval by choosing a sample value from each and substituting back into the simplified inequality to check if the expression is less than zero. Use these results to write the solution set in interval notation, remembering to exclude points where the denominator is zero.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares this ratio to a number or another expression. Solving them requires finding values of the variable that make the inequality true, considering where the expression is defined and the sign of the numerator and denominator.

Critical Points and Sign Analysis

Critical points are values where the numerator or denominator equals zero, causing the rational expression to be zero or undefined. These points divide the number line into intervals, and testing each interval helps determine where the inequality holds by analyzing the sign of the expression in those intervals.

Interval Notation

Interval notation is a concise way to represent sets of real numbers that satisfy inequalities. It uses parentheses for values not included (like points where the expression is undefined) and brackets for values included (like zeros of the numerator when inequality is ≤ or ≥), clearly showing the solution set on the number line.