To rewrite the expression [tex]\(|\pi - 4|\)[/tex] without using absolute value symbols, we need to understand the concept of absolute value.
The absolute value of a number is its distance from 0 on the number line, regardless of the direction. For any real numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex],
[tex]\[ |a - b| = \begin{cases}
a - b & \text{if } a - b \geq 0 \\
b - a & \text{if } a - b < 0
\end{cases} \][/tex]
In our expression, we have [tex]\(\pi\)[/tex] (approximately 3.14159) and 4:
1. First, calculate [tex]\(\pi - 4\)[/tex]:
- [tex]\( \pi \approx 3.14159 \)[/tex]
- So, [tex]\( \pi - 4 \approx 3.14159 - 4 = -0.85841 \)[/tex]
2. Since [tex]\(\pi - 4\)[/tex] is less than 0, the absolute value expression [tex]\(|\pi - 4|\)[/tex] can be rewritten by negating the inside of the absolute value:
[tex]\[
|\pi - 4| = -( \pi - 4)
\][/tex]
3. Simplifying the expression inside the parentheses:
[tex]\[
|\pi - 4| = -(\pi - 4) = 4 - \pi
\][/tex]
So, the expression [tex]\(|\pi - 4|\)[/tex] rewritten without using absolute value symbols is [tex]\(4 - \pi\)[/tex].
Given the result, we can verify:
[tex]\[
4 - \pi \approx 4 - 3.14159 = 0.85841
\][/tex]
Thus, the result from calculating [tex]\(4 - \pi\)[/tex] is approximately [tex]\(0.8584073464102069\)[/tex], confirming the correctness of our rewritten expression.
