Solve:
$\left|5 \frac{2}{3}\right|$
We are asked to evaluate the expression \(\left| 5 \frac{2}{3} \right|\). The term \(5 \frac{2}{3}\) is a mixed number, which signifies the sum of an integer and a fraction:
\[5 \frac{2}{3} = 5 + \frac{2}{3}\]
Since both \(5\) and \(\frac{2}{3}\) are positive, their sum is clearly positive:
\[5 + \frac{2}{3} > 0\]
The absolute value function, denoted by \(|x|\), returns the non-negative value of a real number \(x\). Specifically, if \(x \geq 0\), then \(|x| = x\). Since \(5 \frac{2}{3}\) is positive, its absolute value is simply itself.
To express \(5 \frac{2}{3}\) as an improper fraction, we convert the whole number part to a fraction with the same denominator as the fractional part:
\[5 = \frac{5 \times 3}{3} = \frac{15}{3}\]
Now, we add the fractional part:
\[5 \frac{2}{3} = \frac{15}{3} + \frac{2}{3} = \frac{15 + 2}{3} = \frac{17}{3}\]
Thus, the value of the expression is \(\frac{17}{3}\).
By applying the property of the absolute value function identified in the first step, we conclude that:
\[\left| 5 \frac{2}{3} \right| = 5 \frac{2}{3} = \frac{17}{3}\]
The final result can be expressed as either a mixed number or an improper fraction. Therefore, \(\left| 5 \frac{2}{3} \right| = \boxed{\frac{17}{3}}\).
The concept of absolute value is essential when measuring physical quantities where direction doesn't matter. For example, if a robotic arm is instructed to move \(5 \frac{2}{3}\) inches, the absolute displacement is \(5 \frac{2}{3}\) inches regardless of whether it moves forward or backward from its starting point. Absolute value ensures we focus on the magnitude of the movement, which is critical in engineering, navigation, and everyday measurements.