Question

solve the equation, and enter the solutions from least to greatest. If there is only one solution, enter “n.a” for the second solution. (picture of equation listed below)

Answer 1

x = 1 or x = n.a.

Explanation

`(1)/(x)+(1)/(x-10)=(x-9)/(x-10)`

Multiplying by (x - 10) at both sides of the equation:

`\begin{gathered} (x-10)((1)/(x)+(1)/(x-10))=(x-9)/(x-10)(x-10) \\ \text{ Distributing and simplifying:} \\ (x-10)/(x)+(x-10)/(x-10)=x-9 \\ (x-10)/(x)+1=x-9 \end{gathered}`

Multiplying by x at both sides of the equation:

`\begin{gathered} x((x-10)/(x)+1)=x(x-9) \\ \text{ Distributing and simplifying:} \\ (x(x-10))/(x)+x=x^2-9x \\ x-10+x=x^2-9x \\ 2x-10=x^2-9x \end{gathered}`

Subtracting 2x and adding 10 at both sides of the equation:

`\begin{gathered} 2x-10-2x+10=x^2-9x-2x+10 \\ 0=x^2-11x+10 \end{gathered}`

We can solve this equation with the help of the quadratic formula with the coefficients a = 1, b = -11, and c = 10, as follows:

The solution x = 10 is not possible because it makes zero the denominator in 2 of the rational expressions of the original equation. In consequence, it must be discarded.