3.3 Subtract Integers

Subtraction of Integers

Subtraction with number lines

A number line gives you a visual way to see what happens when you subtract integers.

To subtract on a number line:

1. Start at the minuend (the first number in the problem).
2. If the subtrahend (the second number) is positive, move left that many spaces.
3. If the subtrahend is negative, move right that many spaces.
4. Where you land is the difference.

For example, to solve $5 - 3$, start at 5 and move 3 spaces left. You land on 2. To solve $2 - (-4)$, start at 2 and move 4 spaces right. You land on 6.

Counter method: You can also use two colors of counters (say, yellow for positive and red for negative).

• Start with counters equal to the first number. Use positive counters if it's positive, negative counters if it's negative.
• To subtract a positive number, remove that many positive counters. If you don't have enough, add zero pairs (one positive + one negative) until you do, then remove.
• To subtract a negative number, remove that many negative counters, adding zero pairs if needed.
• Count what's left to find the difference.

Simplification of integer expressions

The single most useful rule for subtracting integers: change subtraction to addition of the opposite. Every subtraction problem can be rewritten as an addition problem.

• Subtracting a positive is the same as adding a negative: $a - b = a + (-b)$ Example: $9 - 5 = 9 + (-5) = 4$

• Subtracting a negative is the same as adding a positive: $a - (-b) = a + b$ Example: $4 - (-6) = 4 + 6 = 10$

Once you convert to addition, use the addition rules you already know:

• Same signs: Add the absolute values and keep the shared sign. $-9 - 2 = -9 + (-2) = -11$

• Different signs: Find the difference of the absolute values and take the sign of the number with the larger absolute value. $-7 - (-3) = -7 + 3 = -4$ (since $|-7| > |3|$, the answer is negative)

Evaluation of variable expressions (algebraic expressions)

To evaluate an expression with variables, plug in the given values and simplify.

Steps:

1. Substitute each variable with its given value (use parentheses around negative numbers).
2. Follow order of operations (PEMDAS).
3. Apply the subtraction-to-addition rule where needed, then simplify.

Example: If $x = 3$ and $y = -5$, evaluate $2x - 3y$.

1. Substitute: $2(3) - 3(-5)$

2. Multiply first: $6 - (-15)$

3. Convert subtraction to addition: $6 + 15 = 21$

Word problems to algebraic expressions

When you see a word problem involving integers, translate it step by step:

1. Identify what quantity you're solving for and assign it a variable.
2. Use positive integers for amounts gained, earned, or increased.
3. Use negative integers (or subtraction) for amounts spent, lost, or decreased.
4. Write the expression and solve.

Example: "John has 10 dollars. He spends 7 dollars on lunch and earns 5 dollars helping a friend. How much does he have now?"

• Let $x$ = John's final amount
• $x = 10 - 7 + 5$
• $x = 8$, so John has 8 dollars

Real-world applications of subtraction

Integer subtraction shows up in everyday situations:

• Temperature changes: If the temperature is $-4°C$ and drops by $6°C$, you calculate $-4 - 6 = -10°C$. If it "drops by $-3°C$" (a double negative), the temperature actually rises: $-4 - (-3) = -4 + 3 = -1°C$.
• Elevation changes: A hiker at 200 feet above sea level descends 350 feet: $200 - 350 = -150$, meaning 150 feet below sea level.
• Financial transactions: A bank balance of 50 dollars minus a 75-dollar charge: $50 - 75 = -25$, meaning 25 dollars overdrawn.

Key Concepts in Integer Subtraction

• Integer: A whole number that can be positive, negative, or zero.
• Additive inverse: The number you add to get zero. The additive inverse of 5 is $-5$ because $5 + (-5) = 0$.
• Inverse operation: Subtraction is the inverse of addition. This means you can "undo" addition with subtraction and vice versa, which is exactly why converting subtraction to addition of the opposite works.