Solve each rational inequality. Give the solution set in interval notation. (a) (x-1)/(x-2) ≥0 (

step 1 In interval notation, we want to solve a variety of inequalities, but they all involve x minus 1

Solve each rational inequality. Give the solution set in...

Key Concepts

Rational Inequalities

A rational inequality is an inequality that involves a rational expression, which is the ratio of two polynomials. Solving such inequalities requires determining where the expression is positive, negative, or zero while considering constraints imposed by the denominator. This concept extends the idea of solving polynomial inequalities by incorporating the additional rules that govern division and undefined expressions.

Critical Points

Critical points are the values of the variable that make either the numerator or the denominator zero. These points divide the number line into intervals within which the sign of the rational expression remains constant. Identifying and properly handling these points is key because they may be zeros of the inequality or points of discontinuity.

Domain Restrictions

Domain restrictions arise from values that make the denominator zero, causing the rational expression to be undefined. These restrictions must be taken into account when solving the inequality because they define the boundaries of the solution set and are not included in the final answer.

Interval Testing

Interval testing involves choosing sample points from the intervals created by the critical points to determine the sign of the rational expression within each interval. This technique helps in deciding which intervals satisfy the original inequality, especially when the inequality includes strict or non-strict comparisons.

Interval Notation

Interval notation is a mathematical shorthand used to represent sets of numbers on the real line. It efficiently communicates the range of solutions that satisfy an inequality, indicating whether endpoints are included or excluded based on the type of inequality (strict or inclusive).

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