Find all real solutions. Check your results. (1)/(x+2)+(1)/(x)=1

Name: Problem 11, Chapter 4: College Algebra with Modeling and Visualization
Uploaded: 2021-06-23T19:00:05-08:00
Duration: 2 min 7 s
Channel: Lauren Shelton
Description: Find all real solutions. Check your results. (1)/(x+2)+(1)/(x)=1

step 1 Okay, so here I have another rational expression, and I see that I have denominators here too. O

Find all real solutions. Check your results....

Key Concepts

Quadratic Equation Solving

After clearing fractions using the LCD, the resulting equation often simplifies to a quadratic form. Solving quadratic equations involves techniques such as factoring, completing the square, or using the quadratic formula. Understanding these methods is essential for finding all potential solutions.

Domain Restrictions

Since denominators in rational expressions cannot be zero, it is crucial to identify and exclude any values that make a denominator vanish. This step protects against including invalid, or extraneous, solutions that may arise during the solution process.

Verification of Solutions

Verifying potential solutions by substituting them back into the original equation is a critical step. This process helps ensure that the solutions do not violate any domain restrictions and confirms that each solution is valid given the original rational equation.

Rational Equations

Rational equations involve fractions whose numerators and denominators are polynomials. Solving them requires careful manipulation, particularly when dealing with terms that have variable expressions in the denominator. This type of equation often needs additional steps compared to linear or quadratic equations due to the presence of these variable denominators.

Least Common Denominator (LCD)

When dealing with equations containing multiple rational expressions, finding the least common denominator is important. The LCD allows one to combine or eliminate fractions in the equation, thereby converting the rational equation into a polynomial equation that is easier to solve.

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