(x-7)(x-1)(2+x)/6x-6=0

Jasonbradley

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Jun 17, 2024

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Solving the Equation: (x-7)(x-1)(2+x)/6x-6=0

This article will guide you through solving the equation (x-7)(x-1)(2+x)/6x-6=0. We'll break down the steps and explain the concepts involved.

Understanding the Equation

The equation (x-7)(x-1)(2+x)/6x-6=0 is a rational equation. This means it involves fractions where the numerator and denominator are polynomials. To solve for x, we need to find the values of x that make the equation true.

Solving the Equation

1. Find the common denominator: The denominator of the fraction is 6x-6. To get rid of the fraction, we can multiply both sides of the equation by 6x-6: (x-7)(x-1)(2+x) = 0

2. Factor the expression: The expression (x-7)(x-1)(2+x) is already factored.

3. Set each factor equal to zero: For the product of factors to be zero, at least one of the factors must be equal to zero:

   • x - 7 = 0
   • x - 1 = 0
   • 2 + x = 0

4. Solve for x:

   • x = 7
   • x = 1
   • x = -2

Checking for Extraneous Solutions

It's important to check our solutions to make sure they don't make the denominator of the original equation equal to zero. If they do, they are called extraneous solutions and are not valid solutions to the equation.

• x = 7: The denominator 6x-6 becomes 6(7)-6 = 36. This is not zero. Therefore, x = 7 is a valid solution.
• x = 1: The denominator 6x-6 becomes 6(1)-6 = 0. This is zero. Therefore, x = 1 is not a valid solution.
• x = -2: The denominator 6x-6 becomes 6(-2)-6 = -18. This is not zero. Therefore, x = -2 is a valid solution.

Solution

The solutions to the equation (x-7)(x-1)(2+x)/6x-6=0 are x = 7 and x = -2.