Question

Solve each equation. $$ \frac{2 x-3}{x+1}=2-\frac{3 x-1}{x+1} $$

   Solve each equation.
$$
\frac{2 x-3}{x+1}=2-\frac{3 x-1}{x+1}
$$

Precalculus

Raymond A. Barnett,… 7th Edition

Chapter 1, Problem 30

Instant Answer

Solved by Expert Nick Johnson

07/17/2021

Step 1

This gives us: $$ 2x - 3 = 2(x+1) - (3x - 1) $$ Show more…

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Solve each equation. $$ \frac{2 x-3}{x+1}=2-\frac{3 x-1}{x+1} $$

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Key Concepts

Rational Equations

Rational equations are equations that involve rational expressions—fractions where the numerator and the denominator are polynomials. Solving these equations generally requires careful attention to the structure of the fractions, as operations like adding, subtracting, or equating the fractions can introduce complexities not seen in simpler linear or quadratic equations.

Domain Restrictions

When dealing with rational expressions, it is crucial to determine the domain of the variable. This involves identifying values that make any denominator equal to zero, which must be excluded from the solution set to avoid undefined expressions. Recognizing these restrictions helps prevent incorrect or extraneous solutions.

Clearing Denominators

A common strategy in solving rational equations is to eliminate the fractions by multiplying both sides of the equation by the least common denominator (LCD). This process simplifies the equation into a polynomial form that is often easier to handle. However, caution must be taken to ensure that any restrictions on the variable are still observed.

Checking for Extraneous Solutions

Extraneous solutions are results that may arise from the algebraic manipulation of an equation, especially when both sides are multiplied by expressions containing variables. It is important to substitute back any potential solutions into the original equation to confirm they do not violate the domain restrictions or lead to mathematical inconsistencies.

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