# If abs(x-2)<6, then what are the bounds of x? i.e. find a and b such that a<x<b.

Jan 21, 2017

$a = - 4$ and $b = 8$

#### Explanation:

$\left\mid x - 2 \right\mid < 6$ if and only if $- 6 < x - 2 < 6$

if and only if $- 6 + 2 < x - 2 + 2 < 6 + 2$

if and only if $- 4 < x < 8$.

A graph often helps to visualise inequalities:
graph{(y-|x-2|)(y-6)=0 [-6.5, 13.5, -1.4, 8.6]}

$\left\mid x - 2 \right\mid < 6$ is the region below the line $y = 6$ which you can see is the interval $- 4 < x < 8$.

Jan 21, 2017

$\left\mid x - 2 \right\mid < 6 \text{ "<=>" ""-"6< x-2<6" "<=>" ""-} 4 < x < 8$.

$a = \text{-} 4 , b = 8.$

#### Explanation:

When an absolute value inequality has an upper limit (like $\left\mid x - 2 \right\mid < 6$ does), what it means is that the magnitude of $x - 2$ needs to be under 6; in other words, $x - 2$ can be positive or negative, but it must be within 6 units of 0, either way.

This means $x - 2$ has a lower limit of $- 6$, but also an upper limit of $6$. We can then rewrite $\left\mid x - 2 \right\mid < 6$ as

$\text{-} 6 < x - 2 < 6$.

The final step is to isolate $x$. We do this by adding 2 to all "sides" of the inequality, to get

$\text{-} 6 + \textcolor{red}{2} < x - 2 + \textcolor{red}{2} < 6 + \textcolor{red}{2}$

or

$\text{-} 4 < x < 8$.

Thus, the lower limit of $x$ is $a = \text{-} 4$, and the upper limit of $x$ is $b = 8$.