Chapter 11.5, Problem 15P

Solve each equation. Check your solutions. If there is no solution, write no solution.

$\frac{x-1}{x+2}+\frac{4x}{2x^2-2x-12}=2$

To determine

To solve the equation and check the solution.

Answer to Problem 15P

The solution is $x=\pm \sqrt{15}$ .

Explanation of Solution

Given information:

The given equation is:

  $\frac{x-1}{x+2}+\frac{4x}{2x^2-2x-12}=2$

Calculation:

The solution can be obtained by:

  $\begin{array}{l}\frac{x-1}{x+2}+\frac{4x}{2x^2-2x-12}=2\\ \frac{x-1}{x+2}+\frac{4x}{2\left(x^2-x-6\right)}=2\\ \frac{x-1}{x+2}+\frac{2x}{x^2-3x+2x-6}=2\\ \frac{x-1}{x+2}+\frac{2x}{x\left(x-3\right)+2\left(x-3\right)}=2\\ \frac{x-1}{x+2}+\frac{2x}{\left(x-3\right)\left(x+2\right)}=2\\ \frac{\left(x-1\right)\left(x-3\right)+2x}{\left(x-3\right)\left(x+2\right)}=2\\ \frac{x^2-3x-x+3+2x}{x^2-x-6}=2\\ x^2-2x+3=2\left(x^2-x-6\right)\\ x^2-2x+3=2x^2-2x-12\\ 2x^2-2x-12-x^2+2x-3=0\\ x^2-15=0\\ x=\pm \sqrt{15}\end{array}$

Hence, the solution is $x=\pm \sqrt{15}$ .

Put the value of $x$ in the solution to check the solution.