- The absolute value \(|x|\) denotes the distance of a point \(x\) from \(0\) on a number line.
- Since both \(|-4|\) and \(|4|\) equal \(4\), the first box represents the value \(4\).
- The definition of absolute value \(|a|\) is the distance from \(a\) to \(0\), so the second box is \(0\).
- Final answer: The first box is \(\boxed{4}\) and the second box is \(\boxed{0}\).
Explanation
- Analyzing Absolute Values and Distance
The absolute value of a number, denoted as \(|x|\), represents its distance from zero on a number line. For any real number \(x\), the distance is always non-negative. Specifically, \(|-4|\) represents the distance from \(-4\) to \(0\), which is \(4\) units. Similarly, \(|4|\) represents the distance from \(4\) to \(0\), which is also \(4\) units. Thus, we have \(|-4| = 4\) and \(|4| = 4\), which confirms that \(|-4| = |4|\).
- Completing the Statements
The first statement is "\(|-4|\) \(\boxed{\phantom{00}}\) is equal to \(|4|\)." Based on the calculation, \(|-4| = 4\) and \(|4| = 4\). The value \(4\) is the result of both expressions, making them equal. Therefore, the value \(4\) correctly completes the relationship within the context of the evaluation. For the second statement, "\(|-4|\) is the distance from \(-4\) to \(\boxed{\phantom{00}}\) on a number line," the definition of absolute value states that the distance from a point \(a\) to \(0\) is \(|a|\). Thus, \(|-4|\) is the distance from \(-4\) to \(0\). The box in this statement should be filled with \(0\).
Examples
Absolute value is a fundamental concept in daily life for measuring differences where direction doesn't matter. For example, if you are tracking your bank account balance, whether you are \(\$5\) above or \(\$5\) below zero, the absolute distance from a balanced account of \(\$0\) is the same: \(\$5\). Understanding that \(|-5| = |5| = 5\) helps in calculating the absolute change in balances or temperatures regardless of whether the change is positive or negative. This is essential for budgeting and monitoring environmental shifts.