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Solving
Solve the rational equation by applying the five-step strategy for equations with rational expressions. This example demonstrates a case where the only algebraic solution is extraneous, resulting in no valid solution for the equation.
Step 1 — Identify restricted values. Factor the quadratic denominator: . Setting each factor equal to zero gives and . Record and .
Step 2 — Find the LCD. The factored denominators are , , and . The LCD is .
Step 3 — Clear the fractions. Multiply every term on both sides by the LCD and cancel matching denominator factors:
Simplify each term: on the left, the entire denominator cancels, leaving . On the right, the first term becomes , and the second term becomes :
Step 4 — Solve the resulting equation. Distribute on the right: . Combine like terms: . Subtract from both sides: . Subtract from both sides: , so .
Step 5 — Check. The algebraic solution matches one of the restricted values recorded in Step 1. Substituting would make the denominators and equal zero, which is undefined. Therefore, is an extraneous solution and must be discarded.
Because the only candidate algebraic solution is extraneous, the original equation has no solution.
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Intermediate Algebra @ OpenStax
Ch.7 Rational Expressions and Functions - Intermediate Algebra @ OpenStax
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Related
A technician is solving an equation to calculate electrical resistance. During the process, they find a numerical result that, when plugged back into the original formula, causes a division by zero. What is the specific term for this type of invalid result?
In a workplace scenario where a technician uses formulas containing fractions with variables in the denominator, a value that is derived algebraically but results in a zero denominator in the original formula is known as an ____ solution.
Match each term related to formula analysis in a technical workplace with its correct definition.
In a professional technical calculation, an algebraically derived value that results in a denominator of zero in the original rational equation is considered a valid solution.
Validating Solutions in Professional Formulas
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Defining Extraneous Solutions in Professional Manuals
A logistics analyst is using a rational equation to calculate the time required for a transport ship to complete its route. After performing the algebraic steps, the analyst identifies a solution that is classified as 'extraneous.' According to standard mathematical validation, what is the defining reason this value must be excluded?
A technician is following a standard operating procedure to verify the results of a rational formula. According to the manual, which specific part of the original equation must be checked to identify and discard an extraneous solution?
Solving
Solving
Solving
Learn After
During a technical audit of a mathematical model, a technician is required to perform the first step of the five-step strategy to solve the following equation: . According to this strategy, which values of must be recorded as restricted values because they would result in a denominator of zero?
A quality control technician is documenting the standard operating procedure for verifying the following mathematical model used in project cost estimations: . Arrange the five steps of the solution strategy in the correct order as they should be applied to this equation.
A manufacturing firm uses the rational equation to model the break-even point for a new product line. Match each part of the solution process with its correct technical designation.
A process engineer is auditing a system model represented by the rational equation . After completing the five-step strategy, the engineer finds that the only potential algebraic solution is . However, because is a restricted value that makes the original denominators equal to zero, it is technically classified as a(n) ____ solution.
A financial analyst is reviewing a budget allocation model defined by the equation . After clearing the fractions and solving the resulting algebraic equation, the calculated value of is a valid operational parameter for this model.