Adding Two Negative Numbers

Calculate $(-4) + (-7)$

Identify the signs: Both numbers are negative = Same signs

Add the absolute values: $|-4| + |-7| = 4 + 7 = 11$ = Sum is 11

Apply the common sign: Both negative, so result is negative = $-11$

Answer: $(-4) + (-7) = -11$

Adding Integers with Different Signs

Calculate $(-9) + 5$

Identify the signs: $-9$ is negative, $5$ is positive = Different signs

Find absolute values: $|-9| = 9$ and $|5| = 5$ = 9 and 5

Subtract smaller from larger: $9 - 5 = 4$ = Difference is 4

Use sign of larger absolute value: 9 > 5, and $-9$ is negative = $-4$

Answer: $(-9) + 5 = -4$

Subtracting a Positive Number

Calculate $3 - 8$

Rewrite as addition: $3 - 8 = 3 + (-8)$ = Adding the opposite

Identify the signs: $3$ is positive, $-8$ is negative = Different signs

Subtract absolute values: $|8| - |3| = 8 - 3 = 5$ = Difference is 5

Apply sign of larger: 8 > 3, and $-8$ is negative = $-5$

Answer: $3 - 8 = -5$

Subtracting a Negative Number

Calculate $(-2) - (-9)$

Rewrite as addition: $(-2) - (-9) = (-2) + 9$ = Subtracting negative = adding positive

Identify the signs: $-2$ is negative, $9$ is positive = Different signs

Subtract absolute values: $|9| - |-2| = 9 - 2 = 7$ = Difference is 7

Apply sign of larger: 9 > 2, and $9$ is positive = $7$

Answer: $(-2) - (-9) = 7$

Real-World: Temperature Change

The temperature at 6 AM was $-7°C$. By noon, it rose 12 degrees. What is the noon temperature?

Set up the expression: Start: $-7$, Change: $+12$ = $(-7) + 12$

Identify the signs: Different signs ($-7$ and $+12$) = Different signs

Subtract absolute values: $12 - 7 = 5$ = Difference is 5

Apply sign of larger: $12 > 7$, positive wins = $5°C$

Answer: The noon temperature is $5°C$

Mistake: Thinking $5 - (-3) = 2$ instead of $8$

Why: Forgetting that subtracting a negative means adding: two negatives make a positive.

Correct: $5 - (-3) = 5 + 3 = 8$. Subtracting $-3$ is the same as adding $3$.

Mistake: Writing $(-4) + (-6) = -2$ instead of $-10$

Why: Mistakenly subtracting instead of adding when both numbers are negative.

Correct: Same signs means ADD absolute values: $4 + 6 = 10$, then keep the negative: $-10$.

Mistake: Getting the wrong sign when subtracting like $3 - 8 = 5$

Why: Forgetting to check which number has the larger absolute value.

Correct: $3 - 8 = 3 + (-8) = -5$ because $8 > 3$ and we keep the negative sign.

Mistake: Confusing $-5 + 3$ with $-5 - 3$

Why: Not carefully reading the operation sign.

Correct: $-5 + 3 = -2$ (moving right on number line), but $-5 - 3 = -8$ (moving left).

Banking Transactions

Banks use integer operations to track deposits (positive) and withdrawals (negative).

If your account has 120 euros and you withdraw 85 euros, then deposit 40 euros: $120 + (-85) + 40 = 75$ euros.

Altitude and Depth

Pilots and divers use integers to track altitude above sea level (positive) or depth below (negative).

A submarine at $-80$ meters dives 25 meters deeper: $-80 + (-25) = -105$ meters.

Game Scores

Many games track gains and losses as positive and negative points.

In a trivia game, you earn 50 points, lose 30, then earn 45: $50 + (-30) + 45 = 65$ points.

Q: Why does subtracting a negative give a positive?

A: Think of it as removing a debt. If you owe 5 dollars ($-5$) and someone cancels that debt (subtracts $-5$), you gain 5 dollars. $0 - (-5) = 0 + 5 = 5$. Subtracting negative = adding positive!

Q: How do I know which sign the answer will have?

A: For addition with different signs, the answer has the sign of the number with the larger absolute value. $-8 + 5 = -3$ because $|-8| = 8 > 5 = |5|$, so the answer is negative.

Q: Is there a quick trick for subtracting integers?

A: Yes! Change subtraction to addition and flip the sign of the second number. Then follow the addition rules. Example: $7 - (-3)$ becomes $7 + 3 = 10$.