I know the answer to this question is: -1. I got that answer by substituting -1 in for x. Please show me how I should do this the correct way.
(x/(3x+6))-((x+1)/(3x-6))=(1/((x^2)-4))
Thank you.
something wrong with my answer two hours ago? Here it is again.
x/(3x+6) - (x+1)/(3x-6) = 1/(x^2-4)
factor out the 1/3 on the left side, and factor the right side
1/3 (x/(x+2) - (x+1)/(x-2)) = 1/((x+2)(x-2))
now put LS over a common denominator
1/3 (x(x-2) - (x+1)(x+2))/((x-2)(x+2)) = 1/((x+2)(x-2))
multiply through by 3(x-2)(x+2)
x(x-2) - (x+1)(x+2) = 3
x^2 - 2x - x^2 - 3x - 2 = 3
-5x - 2 = 3
-5x = 5
x = 1
Steve, thank you for showing me again. Yes, I couldn't quite get it last time. I understood factoring out the 1/3 on the left side. I understood factoring the denominator on the right side. After that I have trouble. Could you please show it to me again with more steps in place?
To find the correct solution for the equation
(x/(3x+6))-((x+1)/(3x-6))=(1/((x^2)-4)),
we will follow these steps:
Step 1: Simplify the expression on the right side of the equation.
The right side of the equation contains 1/((x^2)-4). Notice that ((x^2)-4) is a difference of squares, so we can rewrite it as (x+2)(x-2). Thus, the equation becomes:
(x/(3x+6))-((x+1)/(3x-6)) = 1/((x+2)(x-2))
Step 2: Find a common denominator for the two fractions on the left side of the equation.
The common denominator for the fractions (3x+6) and (3x-6) is (3x+6)(3x-6), which simplifies to (9x^2-36).
Step 3: Multiply each term in the equation by the common denominator.
(x*(3x-6)/(3x+6)(3x-6))-((x+1)(3x+6)/(3x+6)(3x-6)) = 1/((x+2)(x-2))(3x+6)(3x-6)
Simplifying the terms, we get:
(x(3x-6) - (x+1)(3x+6))/ (9x^2-36) = 1/((x+2)(x-2))(3x+6)(3x-6)
Step 4: Expand and simplify the terms on both sides of the equation.
Now, let's expand and simplify the numerator on the left side of the equation:
(x(3x-6) - (x+1)(3x+6)) = -3x^2 - 9
Substituting the simplified numerator and denominator back into the equation, we have:
(-3x^2 -9)/(9x^2-36) = 1/((x+2)(x-2))(3x+6)(3x-6)
Step 5: Cross-multiply and simplify the equation.
We can cross-multiply the fractions. Multiply the numerator on the left side by the denominator on the right side, and multiply the numerator on the right side by the denominator on the left side:
(-3x^2 -9) * ((x+2)(x-2)) = (9x^2-36)
Simplify further:
-3(x^3 - 4) = 9x^2 - 36
Step 6: Solve the equation for x.
Expand and rearrange terms:
-3x^3 + 12 = 9x^2 - 36
Move all terms to one side of the equation:
-3x^3 - 9x^2 + 48 = 0
Now, you can solve this cubic equation for x using various methods. One approach is to use numerical methods or a graphing calculator to find the approximate solutions.