Chapter 11.5, Problem 31PPE

To determine

How could you use cross product to solve $\frac{1}{x-2}=\frac{2x-6}{x+6}+1$

Answer to Problem 31PPE

  $x=2$

Explanation of Solution

Given:

The given equation is

  $\frac{1}{x-2}=\frac{2x-6}{x+6}+1$

Concept used :

• Firstly solve both sides left and right seperately.
• After that multiply the numerator of left side to denominator of right side and then taking equal sign in middle and then multiply the numerator of right side to denominator of left side.

Calculations :

  $\begin{array}{c}\frac{1}{x-2}=\frac{2x-6}{x+6}+1\\ \frac{1}{x-2}=\frac{2x-6}{x+6}+1\cdot \frac{\left(x+6\right)}{\left(x+6\right)}\\ \frac{1}{x-2}=\frac{2x-6+x+6}{x+6}\\ \frac{1}{x-2}=\frac{3x}{x+6}\\ 1\cdot \left(x+6\right)=3x\cdot \left(x-2\right)\\ x+6=3x^2-6x\\ 3x^2-x-6x-6=0\\ 3x^2-7x-6=0\\ 3x^2-\left(9-2\right)x-6=0\\ 3x^2-9x+2x-6=0\\ 3x\left(x-3\right)+2\left(x-3\right)=0\\ \left(3x+2\right)\cdot \left(x-3\right)=0\\ 3x+2=0\\ x=-\frac{2}{3}\\ \&x-3=0\\ x=3\end{array}$