Solve each equation. Check each solution. (1)/(8)+(5 x)/(x+2)=(5)/(2)

Name: Problem 44, Chapter 9: Algebra 2
Uploaded: 2019-08-25T02:15:13-08:00
Duration: 50 s
Channel: K B
Description: Solve each equation. Check each solution. (1)/(8)+(5 x)/(x+2)=(5)/(2)

step 1 So in this question, we're going to multiply to get every term in this equation to have 8 times

Solve each equation. Check each solution. (1)/(8)+(5...

Key Concepts

Rational Equations

A rational equation is an equation that involves at least one fraction whose numerator and/or denominator are polynomials. When solving such equations, it is important to recognize the structure, as the presence of variable expressions in the denominators can impose restrictions on the allowed values of the variable.

Extraneous Solutions

Extraneous solutions are answers obtained through the process of solving an equation that do not satisfy the original equation, typically because they cause a denominator to become zero or violate other conditions in the original equation. Thus, after solving a rational equation, one must always substitute potential solutions back into the original equation to ensure their validity.

Least Common Denominator

Multiplying through by the least common denominator (LCD) is a common technique used to eliminate the fractions in a rational equation. The LCD is the smallest expression that contains all the factors present in the denominators and helps transform the rational equation into a polynomial equation that is easier to manipulate.

Linear Equations

Once the rational equation is cleared of its fractions, it often simplifies to a linear equation. Understanding how to solve linear equations—by isolating the variable using operations such as addition, subtraction, multiplication, and division—is crucial for finding the correct solution.

Solution Verification

Solution verification is an essential step in solving equations, especially rational equations. After obtaining potential solutions, one must plug them back into the original equation to check for any inconsistencies or undefined expressions, ensuring the solutions are true and satisfy all the conditions of the original problem.

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