Solve each equation. a. (1)/(x)=(2 x)/(x+1) b. (1)/(x)+(2 x)/(x+1)=1

Name: Problem 85, Chapter 8: Intermediate Algebra
Uploaded: 2020-08-04T16:49:43-08:00
Duration: 3 min 58 s
Channel: Catt Huth
Description: Solve each equation. a. (1)/(x)=(2 x)/(x+1) b. (1)/(x)+(2 x)/(x+1)=1

step 1 In this question, these are similar only in that we're going to need to get rid of the denominat

Solve each equation. a. (1)/(x)=(2 x)/(x+1) b. (1)/(x)+(2...

Key Concepts

Extraneous Solutions

During the process of solving rational equations, it is possible to introduce solutions that do not satisfy the original equation, often due to domain restrictions or operations like cross multiplication. It is important to verify each potential solution in the original equation to ensure that no extraneous solutions are included.

Algebraic Simplification

After eliminating denominators, the resulting equations often require further manipulation, such as expanding expressions, combining like terms, and factoring. Simplifying the algebraic expressions is essential to isolate the variable and determine the correct solutions.

Rational Equations

This concept refers to equations that involve fractions whose numerators and/or denominators contain algebraic expressions. Solving rational equations typically requires finding a common denominator, cross-multiplication, or other algebraic techniques to eliminate the fractions, allowing the equation to be solved using standard algebraic methods.

Domain Restrictions

When dealing with rational equations, it is crucial to identify values that make any denominator equal to zero, as these are not allowed. Such restrictions ensure that the solutions obtained are meaningful within the context of the original expression and help avoid inadvertent inclusion of values that would make the equation undefined.

Cross Multiplication

This method involves multiplying both sides of an equation by the denominators of the fractions to eliminate them. It is particularly useful when the equation consists of a single fraction on each side. Cross multiplication transforms the equation into a simpler, non-fractional form, making further solving more straightforward.

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