Based on material learned earlier in the course. The purpose...

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam.
Solve the inequality: $\frac{x+1}{x-2} \geq 1$.

Key Concepts

Rational Inequalities

Rational inequalities involve expressions that are ratios of polynomials. They require finding the values of the variable that satisfy an inequality involving a rational function. Typically, the process involves rewriting the inequality so that one side is zero, and then analyzing the sign of the entire expression based on the factors in the numerator and denominator.

Domain Restrictions

When dealing with rational expressions, it is essential to consider the domain of the variable, specifically excluding values that make the denominator zero since the expression is undefined there. This step ensures that any solution set respects the inherent limitations of the rational function.

Critical Points and Sign Analysis

Critical points, which are values where either the numerator or the denominator is zero, divide the number line into intervals within which the sign of the rational expression is constant. A sign chart or test intervals method is typically used to determine where the inequality holds. This analysis is critical for isolating the intervals that satisfy the given inequality.

Algebraic Manipulation in Inequalities

Solving inequalities often requires algebraic manipulations such as moving all terms to one side, finding a common denominator, and combining like terms. Special caution is needed when multiplying both sides by expressions containing variables since doing so can impact the direction of the inequality depending on the sign of the multiplier.

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