Solving Rational Equations and Extraneous Solutions

This video tutorial explains how to solve rational equations and check for extraneous solutions. It begins with an introduction to rational equations, followed by a detailed walkthrough of solving two example equations. The process involves eliminating denominators, factoring, and verifying solutions by substituting them back into the original equations. The tutorial emphasizes the importance of checking for extraneous solutions to ensure the validity of the results.

What is the first step in solving a rational equation to check for extraneous solutions?

• Multiply by the denominator

• Divide by the numerator

• Subtract the denominator

• Add a constant to both sides

When solving the equation x^2 - 7x + 12 = 0, what are the two numbers that multiply to 12 and add to -7?

• -3 and -4

• 1 and 12

• 3 and 4

• 2 and 6

How do you verify if a solution is extraneous?

• By multiplying it by zero

• By solving a different equation

• By checking if it satisfies the original equation

• By adding it to the denominator

What is the first step in solving the equation 5/x + 8/(x+3) = 7?

• Multiply by x and x+3

• Subtract 5 from both sides

• Divide by 8

• Add 7 to both sides

What is the result of multiplying 7x by x+3?

• 7x^2 + 21

• 7x^2 + 21x

• 7x^2 + 3

• 7x + 21

What are the potential solutions for the equation 7x^2 + 8x - 15 = 0?

• x = 3 and x = -5/7

• x = 2 and x = -7/15

• x = 4 and x = -3/7

• x = 1 and x = -15/7

How do you simplify complex fractions in the verification process?

• By multiplying by a common denominator

• By subtracting the denominators

• By dividing by a constant

• By adding the numerators

What is the final step in verifying a solution for a rational equation?

• Add the solution to the equation

• Subtract the solution from the equation

• Multiply the solution by zero

• Check if the equation holds true

What is the purpose of checking for extraneous solutions?

• To verify the validity of solutions

• To eliminate all solutions

• To find new solutions

• To simplify the equation

What is the result of 21 divided by 3 in the context of verifying the second solution?

• 5

• 6

• 7

• 8