# [SOLVED]creating an absolute value equation from an inequallity

#### karush

##### Well-known member
if you are given $$\displaystyle -6 \leq x \leq 14$$

from this how do you create an abs equation like $$\displaystyle |x-4| \leq 10$$

k

#### Chris L T521

##### Well-known member
Staff member
Re: creating an abs equation

if you are given $$\displaystyle -6 \leq x \leq 14$$

from this how do you create an abs equation like $$\displaystyle |x-4| \leq 10$$

k
Note that we can express the absolute value in terms of an inequality: $|y|\leq c \iff -c \leq y \leq c$.

Now, note that if we subtract 4 from each piece of $-6\leq x\leq 14$, we get $-10\leq x-4 \leq 10$. Thus, by what I mentioned in the first line, this means that $|x-4|\leq 10$.

Does this clarify things?

#### Plato

##### Well-known member
MHB Math Helper
Re: creating an abs equation

if you are given $$\displaystyle -6 \leq x \leq 14$$
from this how do you create an abs equation like $$\displaystyle |x-4| \leq 10$$
$$a \le x \le b$$ converts to $$\left| {x - \frac{{a + b}}{2}} \right| \le \frac{{b - a}}{2}$$.

Think of the mid-point of $$[a,b]$$ as well as the radius.