solve the following equation check the solution

Question

1/x-9+1/x+9=18/x^2-81
select the correct choice below and if necessary fill in the answer box to complete your choice
A the solution is
B there are infinitely many solutions
C there is no solution

Answer 1

To solve the equation 1/x-9 + 1/x+9 = 18/x^2-81, we need to first find a common denominator for the fractions on the left side. The common denominator in this case would be (x-9)(x+9).

So, the equation becomes:

(x+9)/(x^2-81) + (x-9)/(x^2-81) = 18/(x^2-81)

Simplifying the left side:

2x/(x^2-81) = 18/(x^2-81)

Now, cross multiply:

2x(x^2-81) = 18

2x^3 - 162x = 18

2x^3 - 162x - 18 = 0

Now, let's solve this cubic equation. By trial and error or using a graphing calculator, we find the solutions to be approximately x = -4.47, x = 2.98, and x = 3.15.

Since there are three distinct solutions for the equation, the correct choice is:

B There are infinitely many solutions (False)

However, for specific values of x, we can check the equation to see if it holds true. Let's substitute one of the values:

For x = 3.15:

1/3.15 - 9 + 1/3.15 + 9 = 18/(3.15)^2 - 81
0.317 - 9 + 0.317 + 9 = 18/9.9225 - 81
-8.686 + 9.317 = 1.8153 - 81
0.630 = -79.1847

The equation does not hold true for x = 3.15, so it can be concluded that there is no solution to the original equation. The correct choice should be:

C There is no solution