How do you solve #6 / (x-4) + 9/x = -36/(x^2 - 4x)#?

Question

Savannah Anderson · Accepted Answer

#x="no solution"# Before starting, it is important to note the restrictions in this equation. When the denominator of each fraction is set to not equal #0#, the restrictions are: #x-4!=0color(white)(XXXXX)x!=0color(white)(XXXXX)x^2-4x!=0# #x!=4color(white)(XXXXXXXXXXXXXXX)x(x-4)!=0# #color(white)(XXXXXXXXXXXXXXXXXX)x!=0,4# Thus, the restrictions are #color(red)(|bar(ul(color(white)(a/a)x!=0,4color(white)(a/a)|)))#. Solving the Equation #1#. Start by adding the two fractions on the left side of the equation. #6/(x-4)+9/x=-36/(x^2-4x)# #(6color(orange)x)/(color(orange)x(x-4))+(9color(blue)((x-4)))/(xcolor(blue)((x-4)))=-36/(x^2-4x)# #2#. Simplify. #(6x)/(x^2-4x)+(9x-36)/(x^2-4x)=-36/(x^2-4x)# #(15x-36)/(x^2-4x)=-36/(x^2-4x)# #3#. Multiply both sides by #color(purple)(x^2-4x)# to get rid of the denominators. #color(purple)((x^2-4x))((15x-36)/(x^2-4x))=color(purple)((x^2-4x))(-36/(x^2-4x))# #4#. Simplify. #color(red)cancelcolor(purple)((x^2-4x))((15x-36)/color(red)cancelcolor(black)((x^2-4x)))=color(red)cancelcolor(purple)((x^2-4x))(-36/color(red)cancelcolor(black)((x^2-4x)))# #15x-36=-36# #5#. Solve for #x#. #15x=0# #x=0# However, looking back at the restrictions #(color(red)(x!=0,4))#, #x=0# is not a valid solution. Therefore: #color(green)(|bar(ul(color(white)(a/a)x="no solution"color(white)(a/a)|)))#

Isaiah Anthony · Answer

undefined To solve the equation $\frac{6}{x-4} + \frac{9}{x} = \frac{-36}{x^2 - 4x}$, you can follow these steps:

1. Find a common denominator for the fractions.
2. Combine the fractions on the left-hand side of the equation.
3. Set the resulting expression equal to the fraction on the right-hand side.
4. Simplify the equation.
5. Solve for $x$.
6. Check for extraneous solutions.

After completing these steps, you'll have the solution for $x$.