# How do you solve abs(2x-4)>10?

Mar 28, 2017

$\textcolor{g r e e n}{x < - 3} \textcolor{w h i t e}{\text{XX")orcolor(white)("XX}} \textcolor{g r e e n}{x > 7}$

#### Explanation:

Here are two methods you might use to solve this:

Method 1: Start by squaring both sides
$\left\mid 2 x - 4 \right\mid > 10$

$\textcolor{w h i t e}{\text{XXX}} \rightarrow {\left(2 x - 4\right)}^{2} > {10}^{2}$

$\textcolor{w h i t e}{\text{XXX}} 4 {x}^{2} - 16 x + 16 > 100$

$\textcolor{w h i t e}{\text{XXX}} {x}^{2} - 4 x - 21 > 0$

$\textcolor{w h i t e}{\text{XXX}} \left(x - 7\right) \left(x + 3\right) > 0$

This will be true it both $\left(x - 7\right)$ and $\left(x + 3\right)$ have the same sign.
That is if
$\textcolor{w h i t e}{\text{XXX}} \left(x - 7\right) < 0 \rightarrow x < 7$
$\textcolor{w h i t e}{\text{XXX}}$and
$\textcolor{w h i t e}{\text{XXX}} \left(x + 3\right) < 0 \rightarrow x < - 3$
$\textcolor{w h i t e}{\text{XXXXXXXXX}}$for both to be true: $x < - 3$
or
$\textcolor{w h i t e}{\text{XXX}} \left(x - 7\right) > 0 \rightarrow x > 7$
$\textcolor{w h i t e}{\text{XXX}}$and
$\textcolor{w h i t e}{\text{XXX}} \left(x + 3\right) > 0 \rightarrow x > - 3$
$\textcolor{w h i t e}{\text{XXXXXXXXX}}$for both to be true: $x > 7$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Method 2: Separate the negative and positive versions of $\left(2 x - 4\right)$ immediately
$\left\mid 2 x - 4 \right\mid > 0$

{: ("if " (2x -4) < 0,color(white)("X")orcolor(white)("X"),"if " (2x-4) > 0), (abs(2x-4) > 10 rarr 4-2x > 10,,abs(2x-4) > 10 rarr 2x-4 > 10), (color(white)("XXXXXXXX")rarr -2x > 6,,color(white)("XXXXXXXX")rarr x > 7), (color(white)("XXXXXXXX")rarr x < -3,,) :}

Mar 28, 2017

$x < - 3$ or $x > 7$

#### Explanation:

$| 2 x - 4 | > 10$

If $| 2 x - 4 |$ is negative,

$- \left(2 x - 4\right) > 10$
$\textcolor{w h i t e}{,} - 2 x + 4 > 10$
$\textcolor{w h i t e}{\times x .} - 2 x > 6$
$\textcolor{w h i t e}{\times \times x .} 2 x < - 6$
$\textcolor{w h i t e}{\times \times x . .} x < - 3$

If $| 2 x - 4 |$ is positive,

$2 x - 4 > 10$
$\textcolor{w h i t e}{\times x} 2 x > 14$
$\textcolor{w h i t e}{\times x .} x > 7$

Hence $x < - 3$ or $x > 7$.