Chapter 9, Problem 46SGR

To determine

To solve: The given equation and check your solution.

Answer to Problem 46SGR

  $x=\frac{46}{17}$

Explanation of Solution

Given information:

The given equation is $\frac{6}{x+5}-\frac{3}{x-3}=\frac{6}{x^2+2x-15}$ .

Calculation:

Eliminate the fraction of the given equation by multiplying each side of the equation by the least common denominator (LCD).

  $\begin{array}{l}\frac{6}{x+5}-\frac{3}{x-3}=\frac{6}{x^2+2x-15}\\ \frac{6}{\left(x+5\right)}-\frac{3}{\left(x-3\right)}=\frac{6}{\left(x+5\right)\left(x-3\right)}\end{array}$

The LCD for the terms is $\left(x+5\right)\left(x-3\right)$ .

  $\begin{array}{l}\frac{6}{\left(x+5\right)}-\frac{3}{\left(x-3\right)}=\frac{6}{\left(x+5\right)\left(x-3\right)}\\ \left(x+5\right)\left(x-3\right)\frac{6}{\left(x+5\right)}-\left(x+5\right)\left(x-3\right)\frac{3}{\left(x-3\right)}=\left(x+5\right)\left(x-3\right)\frac{6}{\left(x+5\right)\left(x-3\right)}\text{ (Multiply by LCD)}\\ 6\left(x-3\right)-3\left(x+5\right)=6\text{ (Divide common factors)}\\ 6x-18-3x-15=6\text{ (Multiply)}\\ 3x-33=6\text{ (Simplify)}\\ 3x=6+33\text{ (Add 33 to both sides)}\\ 3x=39\text{ (Simplify)}\\ x=\frac{39}{3}\text{ (Divide both sides by 3)}\\ x=13\text{ (Simplify)}\end{array}$

Check:

  $\begin{array}{l}\frac{6}{x+5}-\frac{3}{x-3}=\frac{6}{x^2+2x-15}\text{ (Original equation)}\\ \frac{6}{13+5}-\frac{3}{13-3}\stackrel{?}{=}\frac{6}{{\left(13\right)}^2+2\cdot 13-15}\text{ (Substitute }x=13\text{)}\\ \frac{6}{18}-\frac{3}{10}\stackrel{?}{=}\frac{6}{180}\\ \frac{1}{3}-\frac{3}{10}\stackrel{?}{=}\frac{1}{30}\\ \frac{10-9}{30}\stackrel{?}{=}\frac{1}{30}\\ \frac{1}{30}=\frac{1}{30}\end{array}$

Hence,

  $x=13$