Solve each inequality algebraically. ((x+5)^2)/(x^2-4) ≥0 |...

Name: Problem 38, Chapter 5: College Algebra
Uploaded: 2020-05-15T16:43:02-08:00
Duration: 1 min 58 s
Channel: Ankit Gupta
Description: Solve each inequality algebraically. ((x+5)^2)/(x^2-4) ≥0

Key Concepts

Multiplicity of Factors

Multiplicity refers to the number of times a particular factor is repeated in the numerator or denominator. In the context of rational inequalities, factors raised to an even power (such as squared factors) do not change sign when passing through the corresponding zero, which can affect the overall sign pattern of the expression. Recognizing the effect of multiplicities is crucial for accurately conducting sign analysis.

Rational Inequalities

Rational inequalities involve expressions in which one polynomial is divided by another, and an inequality relation (such as ?, >, ?, or <) is imposed on the fraction. The solution process typically requires determining where the expression is defined, finding when it equals zero, and analyzing the sign of the expression over the real number line.

Critical Points and Domain Restrictions

In solving rational inequalities, critical points are the values of the variable that make either the numerator or the denominator zero. Points where the numerator is zero are candidates for the solution (depending on the inequality), while those that make the denominator zero are excluded because they render the expression undefined. These critical values partition the number line into intervals for further analysis.

Sign Analysis

Sign analysis is used to determine the positivity or negativity of the rational expression across the intervals determined by the critical points. By evaluating the signs of the numerator and denominator separately, one can infer the sign of the entire expression in each interval, which is essential for solving the inequality.

Test Intervals

After identifying the critical points, the number line is divided into intervals. A test point is chosen from each interval to substitute back into the rational expression. This process allows one to check if the expression meets the inequality condition, ensuring a systematic exploration of every possible interval.

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