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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)

solve the equation, and enter the solutions from least to greatest. If there is only-example-1
User TLGreg
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1 Answer

3 votes

Answer

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:


\begin{gathered} x_(1,2)=\frac{-b\pm{}√(b^2-4ac)}{2a} \\ x_(1,2)=\frac{11\pm\sqrt{(-11)^2-4\cdot1\operatorname{\cdot}10}}{2\operatorname{\cdot}1} \\ x_(1,2)=(11\pm√(81))/(2) \\ x_1=(11+9)/(2)=10 \\ x_2=(11-9)/(2)=1 \end{gathered}

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.

User Sivanesh S
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3.1k points