MHB Creating an absolute value equation from an inequallity

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To create an absolute value equation from the inequality -6 ≤ x ≤ 14, first recognize that this can be expressed in terms of absolute value. By subtracting 4 from each part of the inequality, you derive -10 ≤ x - 4 ≤ 10. This leads to the absolute value equation |x - 4| ≤ 10. Additionally, the general formula for converting an interval a ≤ x ≤ b into absolute value form is |x - (a + b)/2| ≤ (b - a)/2, which reinforces the derived equation. Understanding these transformations clarifies how to represent inequalities using absolute values effectively.
karush
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if you are given $$-6 \leq x \leq 14$$

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

k
 
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Re: creating an abs equation

karush said:
if you are given $$-6 \leq x \leq 14$$

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

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?
 
Re: creating an abs equation

karush said:
if you are given $$-6 \leq x \leq 14$$
from this how do you create an abs equation like $$|x-4| \leq 10$$

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.
 
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