Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (−x+2)/(x−4)≥0

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials set in relation to zero, such as (−x+2)/(x−4)≥0. To solve these inequalities, one must determine where the rational expression is positive or zero, which requires finding critical points where the numerator and denominator equal zero. The solution set is then identified by testing intervals around these critical points.

Critical Points

Critical points are values of the variable that make the numerator or denominator of a rational expression equal to zero. For the inequality (−x+2)/(x−4)≥0, the critical points are x=2 (where the numerator is zero) and x=4 (where the denominator is zero). 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 on the real number line. 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 may be expressed as [2, 4) or (-∞, 2) ∪ (4, ∞), depending on the results of the inequality tests.