Question

Solve the equation both algebraically and graphically. (Lesson 1.11)

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

Short answer

The solution by algebraically is $x=-4$and graphically is given below:

Step-by-step solution

Step 1. Definition of the intersection point.

An intersection is a single point where two lines meet or cross each other.

Step 2. Solve equation algebraically.

We have$\frac{4}{x+2}-\frac{6}{2x}=\frac{5}{2x+4}$

Multiply both sides by $x+2$:

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

Add 4 both sides:

$\begin{array}{l}4-\frac{6\left(x+2\right)}{2x}-4=\frac{5}{2}-4\\ -\frac{6\left(x+2\right)}{2x}=\frac{5-8}{2}\\ -\frac{6\left(x+2\right)}{2x}=\frac{-3}{2}\\ \frac{6\left(x+2\right)}{2x}=\frac{3}{2}\end{array}$

Step 3. Further, calculate.

Multiply both sides by 2x

$\begin{array}{l}2x\left(\frac{6\left(x+2\right)}{2x}\right)=2x\left(\frac{3}{2}\right)\\ 6x+12=3x\\ 6x-3x=-12\\ 3x=-12\\ x=-\frac{12}{3}\\ x=-4\end{array}$

Step 4. Solve the equation graphically.

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

Now, equate the left side of the equation to $y_1$and the right side to $y_2$.

$\begin{array}{l}{y}_1=\frac{4}{x+2}-\frac{6}{2x}\\ y_2=\frac{5}{2x+4}\end{array}$

Draw the graph:

As we can see that both lines are intersecting at $\left(-4,-1.25\right)$.

So, the solution is $x=-4$.