Question

solve the rational equation, State the excluded values -2/x-8 = x-1/x+2

Answer 1

Answer::

$\begin{gathered} x=3\text{ or 4} \\ x\neq8,x\neq-2 \end{gathered}$

Explanation:

Given the rational function:

$\frac{-2}{x-8}=\frac{x-1}{x+2}$

Step 1: Cross-multiply

$-2(x+2)=(x-1)(x-8)$

Step 2: Expand and simplify

$\begin{gathered} -2x-4=x^2-8x-x+8 \\ -2x-4=x^2-9x+8 \\ x^2-9x+2x+8+4=0 \\ x^2-7x+12=0 \end{gathered}$

Step 3: Solve the quadratic equation for x.

$\begin{gathered} x^2-7x+12=0 \\ x^2-4x-3x+12=0 \\ x(x-4)-3(x-4)=0 \\ (x-3)(x-4)=0 \\ x-3=0\text{ or }x-4=0 \\ x=3\text{ or }x=4 \end{gathered}$

Step 4: Find the excluded values

The excluded values are the values at which the function is undefined.

Set the denominators equal to zero.

$\begin{gathered} x-8=0\text{ or x+2=0} \\ x=8,x=-2 \end{gathered}$

Thus the solution of the function is:

$\begin{gathered} x=3\text{ or 4} \\ x\neq8,x\neq-2 \end{gathered}$