Chapter 1.9, Problem 34E

To determine

To calculate: The value of $x$ from the equation $\frac{4}{x+2}-\frac{6}{2x}=\frac{5}{2x+4}$ by algebraic method and graphical method.

Answer to Problem 34E

The value of $x$ is $-\frac{2}{3}$ and Graph of the equation is provided below,

  

Explanation of Solution

Given information:

The equation $\frac{4}{x+2}-\frac{6}{2x}=\frac{5}{2x+4}$ .

Calculation:

It is provided that the equation $\frac{4}{x+2}-\frac{6}{2x}=\frac{5}{2x+4}$ .

Algebraic method;

The method in which isolate $x$ on one side of the equation and solve the value of $x$ .

From the equation $\frac{4}{x+2}-\frac{6}{2x}=\frac{5}{2x+4}$ ,

  $\begin{array}{c}\frac{4}{x+2}-\frac{6}{2x}=\frac{5}{2x+4}\\ \frac{4.2x-6\left(x+2\right)}{2x\left(x+2\right)}=\frac{5}{2\left(x+2\right)}\\ \frac{8x-6x-2}{2x\left(x+2\right)}=\frac{5}{2\left(x+2\right)}\end{array}$

Solve the value of $x$ from the equation,

  $\begin{array}{l}\frac{8x-6x-2}{2x\left(x+2\right)}=\frac{5}{2\left(x+2\right)}\\ \frac{2x-2}{2x\left(x+2\right)}=\frac{5}{2\left(x+2\right)}\\ x=-\frac{2}{3}\end{array}$

Therefore, the value of $x=-\frac{2}{3}$ .

Graphical method;

Let put $y_1=\frac{4}{x+2}-\frac{6}{2x}$ and $y_2=\frac{5}{2x+4}$

The graph of the equation $y_1=\frac{4}{x+2}-\frac{6}{2x}$ provided below,

  

The graph of equation $y_2=\frac{5}{2x+4}$ is provided below,

  

The graph of both equations in same figure is provided below,

  

The solution is the $x$ coordinate of the intersection point $\left(-\frac{2}{3},5\right)$ of the two graphs. So, the intersection point on the $x$ coordinate is $\frac{-2}{3}$ .

Thus, the value of $x$ is $\frac{-2}{3}$ .