Creating an absolute value equation from an inequallity

[karush]

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]

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?

 

[Plato2]

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.