Solve each rational inequality. Give the solution set in interval notation. (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. For the numerator: \(x - 1 = 0\), which gives \(x = 1\). For the denominator: \(x - 6 = 0\), which gives \(x = 6\). These points divide the number line into intervals to test.
2. Determine the intervals to test based on the critical points: \((-\infty, 1)\), \((1, 6)\), and \((6, \infty)\). Remember that \(x = 6\) is excluded from the domain because it makes the denominator zero.
3. Choose a test point from each interval and substitute it into the expression \(\frac{x - 1}{x - 6}\) to check the sign (positive or negative) of the expression in that interval.
4. Since the inequality is \(\frac{x - 1}{x - 6} \leq 0\), include intervals where the expression is negative or zero. Also, check if the expression equals zero at any critical points and include those points if they satisfy the inequality.
5. Combine the intervals where the inequality holds true, excluding any points where the expression is undefined (like \(x = 6\)), and write the solution set in interval notation.

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, exclude values that make the denominator zero, as these are not in the domain, and use brackets for points where the expression equals zero if included.