# How do you solve 1/x-1/(2x)=2x ?

## I know the solution to $\frac{1}{x} - \frac{1}{2 x} = 2 x$ is $x = \pm 0.5$ At least it is the solution in my math book. But I don't understand how you solve it. When I tried i got $x = \pm 0 , 25$ I think. So I typed the problem into Cymath. And it tells me to do the following: One. Simplify $\frac{1}{x} - \frac{1}{2 x}$ to $\frac{1}{2 x}$ $\frac{1}{2 x} = 2 x$ Two. Multiply both sides by $2 x$ $1 = 2 x \cdot 2 x$ Three. Simplify $2 x \cdot 2 x$ to $4 {x}^{2}$ $1 = 4 {x}^{2}$ Four. Divide both sides by $4$ $\frac{1}{4} = {x}^{2}$ Five. Take the square root of both sides. $\pm \sqrt{\frac{1}{4}} = x$ So the answer is $x = \pm 0.5$ But the thing I don't understand is the first step. Why does $\frac{1}{x} - \frac{1}{2 x}$ equals to $\frac{1}{2 x}$? It just seems so illogical to me. Is there another way to solve the equation? Or can someone please explain why this is?

May 9, 2018

#### Explanation:

$\frac{1}{x} - \frac{1}{2 x} = 2 x \implies$ Left side LCD = $2 x$ :

$\frac{2}{2 x} - \frac{1}{2 x} = 2 x \implies$ or:

$\frac{1}{2 x} = 2 x$

Let's continue:

Multiply both sides by $2 x$:

$\frac{2 x}{2 x} = 2 x \cdot 2 x$

$1 = 4 {x}^{2} \implies$ divide both sides by$4$:

$\frac{1}{4} = {x}^{2} \implies$ take square root of both sides:

$\pm \frac{1}{2} = x \implies$ or:

$x = \pm 0.5$

2nd method:

$\frac{1}{x} - \frac{1}{2 x} = 2 x$

To get rid of denominators multiply both sides of equation by the Lowest Common Denominator (LCD) in this case $2 x$:

$\frac{2 x}{x} - \frac{2 x}{2 x} = 2 x \cdot 2 x \implies$ simplify:

$2 - 1 = 4 {x}^{2} \implies$ simplify:

$1 = 4 {x}^{2} \implies$ continue same as above

May 9, 2018

$x = \pm \frac{1}{2}$

#### Explanation:

Given: $\frac{1}{x} - \frac{1}{2 x} = 2 x$

Considering the left hand side only for a moment $\frac{1}{x} - \frac{1}{2 x}$
We need to make the bottom values (denominators) the same. Multiply by 1 and you do not change the value. However, 1 comes in many forms.

$\textcolor{g r e e n}{\left[\frac{1}{x} \textcolor{red}{\times 1}\right] - \frac{1}{2 x} = 2 x \textcolor{w h i t e}{\text{dddd")->color(white)("dddd}} \left[\frac{1}{x} \textcolor{red}{\times \frac{2}{2}}\right] - \frac{1}{2 x} = 2 x}$

color(green)(color(white)("ddddddddddddddddddddd")->color(white)("ddddd")ubrace([2/(2x)]color(white)("dd")-1/(2x))=2x)
$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{ddddddddddddddddddddddddddd-dddddd}} \downarrow}$

$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{ddddddddddddddddddddd")->color(white)("dddddddddd") 1/(2x)color(white)("dddd}} = 2 x}$

Multiply both sides by $\textcolor{red}{x}$

$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{ddddddddddddddddddddd")->color(white)("ddddd") 1/(2cancel(x))color(red)(xx cancel(x))color(white)("dddd}} = 2 x \textcolor{red}{\times x}}$

color(green)(color(white)("ddddddddddddddddddddd")->color(white)("ddddddddd")1/2=2x^2

Divide both sides by 2

color(green)(color(white)("ddddddddddddddddddddd")->color(white)("ddddddddd")1/4=x^2

Square root both sides
color(green)( color(white)("ddddddddddddddddddddd")->color(white)("ddddddd")+-1/2=x
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
$\textcolor{b l u e}{\text{Check}}$

Set $x = + \frac{1}{2}$
$\frac{1}{x} - \frac{1}{2 x} = 2 x$

$\frac{1}{\frac{1}{2}} - \frac{1}{2 \times \frac{1}{2}} = 2 \times \frac{1}{2}$ ?

$2 - 1 = 1 \textcolor{red}{\leftarrow \text{True}}$

Set $x = - \frac{1}{2}$
$\frac{1}{x} - \frac{1}{2 x} = 2 x$

$\frac{1}{- \frac{1}{2}} - \frac{1}{2 \times \left(- \frac{1}{2}\right)} = 2 \times \left(- \frac{1}{2}\right)$ ?

$- 2 + 1 = - 1 \textcolor{red}{\leftarrow \text{True}}$