Solve the rational inequality. (x^2-4)/(x+5) ≥0

Key Concepts

Domain Restrictions

When working with rational expressions, it is crucial to identify values that make the denominator zero, because these are not allowed in the domain of the function. These restrictions ensure that the solution set only includes values where the expression is defined.

Factorization

Factorization is a mathematical technique used to break down polynomials into simpler components (such as linear factors), which makes it easier to identify the zeros of the numerator or denominator. This simplification aids in the efficient determination of critical points and the subsequent sign analysis.

Sign Chart Analysis

Sign chart analysis involves picking test points in the intervals determined by the critical points to check whether the rational expression is positive or negative in those intervals. This step is essential for determining the solution set of the inequality by assessing where the expression meets the inequality condition.

Rational Inequality

A rational inequality involves a fraction where both the numerator and the denominator are polynomial expressions, and the inequality sign (such as ?, ?, >, or <) indicates a range of values for which the expression satisfies the given inequality. Solving such inequalities typically requires analyzing where the rational expression is positive, negative, or zero.

Critical Points Identification

Critical points are the values of the variable that make the numerator zero (which could yield a zero or change the sign of the expression) and the values that make the denominator zero (which are points of discontinuity). Identifying these points allows one to break the number line into intervals for further analysis.

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