- Start by rewriting subtractions of negative numbers as additions of their positive counterparts: \(a - (-b) = a + b\).
- Use the number line by starting at the first number and moving to the right for adding (or subtracting a negative) and to the left for subtracting a positive.
- Calculate the values: \({}^{-}5 - {}^{-}1 = \boxed{{}^{-}4}\), \({}^{-}5 - {}^{-}4 = \boxed{{}^{-}1}\), \(3 - {}^{-}5 = \boxed{8}\), and \({}^{-}2 - 5 = \boxed{{}^{-}7}\).
Explanation
- Analyzing the Expressions
To compute subtractions using a number line, we represent the minuend as a starting point. Subtracting a positive number \(b\) means moving \(b\) units to the left. Subtracting a negative number \(-b\) is equivalent to adding its opposite, \(b\), which means moving \(|b|\) units to the right. We represent the given expressions as follows:
A\({}^{-}5 - {}^{-}1 = {}^{-}5 + 1\)
B\({}^{-}5 - {}^{-}4 = {}^{-}5 + 4\)
C\(3 - {}^{-}5 = 3 + 5\)
D\({}^{-}2 - 5 = {}^{-}2 - 5\)
- Applying the Number Line Model
For each expression, we start at the minuend on the number line and move the specified distance in the appropriate direction:
AFor \({}^{-}5 - {}^{-}1\): Start at \({}^{-}5\). Since we are subtracting a negative, move 1 unit to the right. We land on \({}^{-}4\).
BFor \({}^{-}5 - {}^{-}4\): Start at \({}^{-}5\). Since we are subtracting a negative, move 4 units to the right. We land on \({}^{-}1\).
CFor \(3 - {}^{-}5\): Start at \(3\). Since we are subtracting a negative, move 5 units to the right. We land on \(8\).
DFor \({}^{-}2 - 5\): Start at \({}^{-}2\). Since we are subtracting a positive 5, move 5 units to the left. We land on \({}^{-}7\).
- Final Results
Based on the movements on the number line:
A\({}^{-}5 - {}^{-}1 = \boxed{{}^{-}4}\)
B\({}^{-}5 - {}^{-}4 = \boxed{{}^{-}1}\)
C\(3 - {}^{-}5 = \boxed{8}\)
D\({}^{-}2 - 5 = \boxed{{}^{-}7}\)
Each result is determined by identifying the final coordinate on the number line.
Examples
Understanding how to add or subtract integers is crucial for tracking changes in real-world scenarios. For example, if your bank account balance is -5 dollars and you receive a rebate of 1 dollar, your balance changes to -5 + 1 = -4 dollars. Similarly, if the temperature is 3 degrees and drops by -5 degrees (which is an increase in temperature), the new temperature becomes 3 + 5 = 8 degrees. Mastering these movements on a number line helps you visualize financial transactions, temperature shifts, and other additive processes in daily life.