Solve the rational inequality. (1)/(x-1)+(1)/(x+1)>(3)/(4) |...

Name: Problem 74, Chapter 4: College Algebra with Modeling and Visualization
Uploaded: 2021-09-29T22:21:55-08:00
Duration: 1 min 16 s
Channel: James Kiss
Description: Solve the rational inequality. (1)/(x-1)+(1)/(x+1)>(3)/(4)

Key Concepts

Rational Inequalities

A rational inequality involves fractions where the variables appear in denominators, and solving them requires careful consideration of the signs of these expressions. The main goal is to determine for which values of the variable the entire rational expression meets the inequality condition.

Domain Considerations

When working with rational expressions, it is essential to first determine the domain, i.e., the values that make any denominator zero must be excluded. Recognizing these restrictions ensures that only valid values are considered when solving the inequality.

Common Denominator and Combining Fractions

Combining fractions with different denominators involves finding a common denominator, which simplifies the expression into a single rational expression. This step is crucial in simplifying the inequality to a more workable form.

Multiplication by Variable Expressions

When clearing fractions in inequalities, one often multiplies both sides by an expression that contains the variable. This requires careful handling because if the expression might be negative, the direction of the inequality could change. This step necessitates splitting the problem into cases based on the sign of that expression.

Critical Points and Sign Analysis

After simplifying the inequality, determining the critical points — values that make the numerator or denominator zero — is essential. These points divide the number line into intervals. A sign chart analysis on these intervals helps in identifying where the inequality holds, taking into account the change of sign at these critical points.

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