We are going to solve the rational equation [tex] \frac{8}{x+ \frac{1}{5} } = \frac{3}{x} [/tex]
Step 1. Simplify the denominator of the left hand side of the equation:
[tex]\frac{8}{x+ \frac{1}{5} } = \frac{3}{x} [/tex]
[tex] \frac{8}{ \frac{5x+1}{5} } = \frac{3}{x} [/tex]
[tex] \frac{40}{5x+1} = \frac{3}{x} [/tex]
Step 2. Multiply both sides of the equation by [tex](5x+1)[/tex] and [tex]x[/tex]:
[tex]\frac{40}{5x+1}(5x+1)(x) = \frac{3}{x}(5x+1)(x)[/tex]
[tex]40x=3(5x+1)[/tex]
[tex]40x=15x+3[/tex]
Step 3. Subtract [tex]15x[/tex] from both sides of the equation:
[tex]40x-15x=15x-15x+3[/tex]
[tex]25x=3[/tex]
Step 4. Divide both sides of the equation by 25:
[tex] \frac{25x}{25} = \frac{3}{25} [/tex]
[tex]x= \frac{3}{25} [/tex]
We can conclude that the solution of the rational equation is [tex]x= \frac{3}{25} [/tex]
