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: \(x - 1 = 0\) and \(x - 6 = 0\).
2. Solve these equations to find the critical points: \(x = 1\) and \(x = 6\).
3. Use these critical points to divide the number line into intervals: \((-\infty, 1)\), \((1, 6)\), and \((6, \infty)\).
4. Test a point from each interval in the inequality \(\frac{x-1}{x-6} \leq 0\) to determine where the inequality holds true.
5. Combine the intervals where the inequality is satisfied and express the solution set in interval notation, considering whether the endpoints are included based on the inequality sign.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials set in relation to a value, typically zero. To solve these inequalities, one must determine where the rational expression is positive, negative, or zero. This often requires finding critical points where the numerator or denominator equals zero, which helps in analyzing the sign of the expression across different intervals.

Critical Points

Critical points are values of the variable that make the rational expression undefined or equal to zero. For the inequality (x-1)/(x-6)≤0, the critical points are x=1 (where the expression equals zero) and x=6 (where the expression is undefined). These points divide the number line into intervals that can be tested to determine where the inequality holds true.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses parentheses and brackets to indicate whether endpoints are included (closed intervals) or excluded (open intervals). For example, the solution set for the inequality can be expressed in interval notation to clearly communicate the values of x that satisfy the inequality, such as [1, 6) for values between 1 and 6, including 1 but not 6.