Solve each rational inequality. Give the solution set in interval notation. See Examples 8 and 9. (x-1)/(x-6)≤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 \( x - 6 = 0 \). These points divide the number line into intervals.
2. Determine the intervals based on the critical points: \( (-\infty, 1) \), \( (1, 6) \), and \( (6, \infty) \).
3. Test a value from each interval in the inequality \( \frac{x-1}{x-6} \leq 0 \) to check whether the expression is less than or equal to zero in that interval.
4. Consider the points where the numerator is zero (\( x=1 \)) because the expression equals zero there, so include this point in the solution set if it satisfies the inequality.
5. Exclude the points where the denominator is zero (\( x=6 \)) because the expression is undefined there, and write the solution set in interval notation based on the intervals where the inequality holds true.

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 finding where the expression is positive, negative, or zero, considering the domain restrictions where the denominator is not zero.

Critical Points and Sign Analysis

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

Interval Notation and Domain Restrictions

Interval notation expresses solution sets compactly using parentheses and brackets to indicate open or closed intervals. When solving rational inequalities, it is essential to exclude values that make the denominator zero, as these are not in the domain and cannot be part of the solution.