Solve each rational inequality. Give the solution set in interval notation. (x - 1)/(x - 4) > 0

Verified step by step guidance

1. Identify the critical points by setting the numerator and denominator equal to zero separately: solve \(x - 1 = 0\) and \(x - 4 = 0\). These points divide the number line into intervals.
2. The critical points are \(x = 1\) and \(x = 4\). These points are where the rational expression can change sign or be undefined (note that \(x = 4\) makes the denominator zero, so it is excluded from the domain).
3. Determine the sign of the expression \(\frac{x - 1}{x - 4}\) on each interval created by the critical points: \(( -\infty, 1 )\), \((1, 4)\), and \((4, \infty)\). Choose a test point from each interval and substitute it into the expression to check if the result is positive or negative.
4. Since the inequality is \(\frac{x - 1}{x - 4} > 0\), select the intervals where the expression is positive based on your test points. Remember to exclude \(x = 4\) because the expression is undefined there.
5. Express the solution set in interval notation by combining the intervals where the inequality holds true, using parentheses to indicate that endpoints where the expression is zero or undefined are not included.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions where one polynomial is divided by another, and the inequality compares this ratio to zero or another value. Solving them requires understanding where the expression is positive, negative, or undefined by analyzing the numerator and denominator separately.

Critical Points and Sign Analysis

Critical points are values that make the numerator or denominator zero, dividing the number line into intervals. By testing values in each interval, you determine the sign of the rational expression, which helps identify where the inequality holds true.

Interval Notation

Interval notation is a concise way to represent solution sets on the number line using parentheses and brackets. Parentheses indicate values not included, while brackets include endpoints. It is essential for expressing the final solution of inequalities clearly.