Solved Exercises on Subtracting Integers in Grade 7

Master subtracting integers: positive, negative, zero pairs, absolute value, number line representation through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Positive - Positive
Exercise 1
Calculate: \( (+8) - (+3) \)
Definition:

Subtracting integers: Change subtraction to addition of the opposite: \(a - b = a + (-b)\)

Subtraction method:
  1. Change subtraction to addition of the opposite
  2. \( (+8) - (+3) = (+8) + (-3) \)
  3. Apply addition rules for different signs
Original Expression
\((+8) - (+3)\)
Change to Addition
\((+8) + (-3)\)
Apply Addition Rules
\(|8| > |3|, 8 - 3 = 5\)
Result
\(5\)
Step 1: Change subtraction to addition of the opposite

\( (+8) - (+3) = (+8) + (-3) \)

Step 2: Apply addition rules for different signs

Compare absolute values: \(|+8| = 8\) and \(|-3| = 3\)

Since \(8 > 3\), subtract: \(8 - 3 = 5\)

Step 3: Take the sign of the number with larger absolute value

Since \(|+8| > |-3|\), take the positive sign

Step 4: Write the final answer

\(+5\) or simply \(5\)

\( (+8) - (+3) = 5 \)
Final answer:

\( (+8) - (+3) = 5 \)

Applied rules:

• Subtraction rule: \(a - b = a + (-b)\)

• Different signs: Subtract absolute values, take sign of larger

• Concept: Subtracting a positive is like adding a negative

2 Positive - Negative
Exercise 2
Calculate: \( (+6) - (-4) \)
Definition:

Subtracting a negative: This is equivalent to adding a positive: \(a - (-b) = a + b\)

Original Expression
\((+6) - (-4)\)
Change to Addition
\((+6) + (+4)\)
Apply Addition Rules
\(6 + 4 = 10\)
Result
\(10\)
Step 1: Change subtraction to addition of the opposite

\( (+6) - (-4) = (+6) + (+4) \)

Step 2: Apply addition rules for same signs

Both numbers are positive: add their absolute values

\(|+6| + |+4| = 6 + 4 = 10\)

Step 3: Keep the common positive sign

Since both were positive, the result is positive

Step 4: Write the final answer

\(+10\) or simply \(10\)

\( (+6) - (-4) = 10 \)
Final answer:

\( (+6) - (-4) = 10 \)

Applied rules:

• Subtracting a negative: \(a - (-b) = a + b\)

• Same signs: Add absolute values, keep the common sign

• Concept: Subtracting a negative is like gaining the opposite of a loss

3 Negative - Positive
Exercise 3
Calculate: \( (-7) - (+2) \)
Definition:

Subtracting a positive from a negative: Makes the result more negative

Original Expression
\((-7) - (+2)\)
Change to Addition
\((-7) + (-2)\)
Apply Addition Rules
\(7 + 2 = 9\)
Keep Negative Sign
\(-9\)
Step 1: Change subtraction to addition of the opposite

\( (-7) - (+2) = (-7) + (-2) \)

Step 2: Apply addition rules for same signs

Both numbers are negative: add their absolute values

\(|-7| + |-2| = 7 + 2 = 9\)

Step 3: Keep the common negative sign

Since both were negative, the result is negative: \(-9\)

Step 4: Write the final answer

\(-9\)

\( (-7) - (+2) = -9 \)
Final answer:

\( (-7) - (+2) = -9 \)

Applied rules:

• Subtraction rule: \(a - b = a + (-b)\)

• Same signs: Add absolute values, keep the common sign

• Concept: Subtracting a positive from a negative makes it more negative

Rules and methods, laws,...
\( a - b = a + (-b) \)
Subtraction Rule
Subtract Positive
\( a - (+b) = a + (-b) \)
Subtracting positive = Adding negative
Subtract Negative
\( a - (-b) = a + b \)
Subtracting negative = Adding positive
Change Rule
\( a - b = a + (-b) \)
Always change to addition of opposite
Key definitions:

Integer: Whole number including positive, negative, and zero

Absolute value: Distance from zero on number line, always non-negative

Opposites: Numbers that are the same distance from zero but in opposite directions

Subtraction rule: \(a - b = a + (-b)\) - Change subtraction to addition of the opposite

Integer subtraction methods:
  1. Change rule: Convert subtraction to addition of the opposite
  2. Apply addition: Use previously learned addition rules
  3. Same signs: Add absolute values, keep the common sign
  4. Different signs: Subtract absolute values, take sign of larger
Tip 1: Remember: "Keep-Change-Change" - Keep first number, change subtraction to addition, change sign of second number
Tip 2: Subtracting a negative is the same as adding a positive
Tip 3: Use a number line to visualize when confused
Tip 4: Always change subtraction to addition first
Common errors: Forgetting to change the sign of the number being subtracted, mixing up subtraction/addition rules, not applying the change rule.
Exam preparation: Practice the "Keep-Change-Change" method, memorize the subtraction rule, use number lines for verification.
Formulas to know by heart:

• Subtraction rule: \( a - b = a + (-b) \)

• Subtract positive: \( a - (+b) = a + (-b) \)

• Subtract negative: \( a - (-b) = a + b \)

• Opposite subtraction: \( (+a) - (-a) = +2a \) and \( (-a) - (+a) = -2a \)

Solution: Exercises 4 to 5
4 Negative - Negative
Exercise 4
Calculate: \( (-5) - (-9) \)
Definition:

Subtracting a negative from a negative: Can result in a positive number

Original Expression
\((-5) - (-9)\)
Change to Addition
\((-5) + (+9)\)
Compare Absolute Values
\(9 > 5\)
Subtract
\(9 - 5 = 4\)
Take Sign of Larger
\(4\)
Step 1: Change subtraction to addition of the opposite

\( (-5) - (-9) = (-5) + (+9) \)

Step 2: Apply addition rules for different signs

Compare absolute values: \(|-5| = 5\) and \(|+9| = 9\)

Since \(9 > 5\), subtract: \(9 - 5 = 4\)

Step 3: Take the sign of the number with larger absolute value

Since \(|+9| > |-5|\), take the positive sign

Step 4: Write the final answer

\(+4\) or simply \(4\)

\( (-5) - (-9) = 4 \)
Final answer:

\( (-5) - (-9) = 4 \)

Applied rules:

• Subtraction rule: \(a - (-b) = a + b\)

• Different signs: Subtract absolute values, take sign of larger

• Concept: Subtracting a negative "undoes" the negative effect

5 Subtracting Opposites
Exercise 5
Calculate: \( (+7) - (-7) \)
Definition:

Subtracting opposites: Results in twice the absolute value of the number

Original Expression
\((+7) - (-7)\)
Change to Addition
\((+7) + (+7)\)
Add Absolute Values
\(7 + 7 = 14\)
Keep Positive Sign
\(14\)
Step 1: Change subtraction to addition of the opposite

\( (+7) - (-7) = (+7) + (+7) \)

Step 2: Apply addition rules for same signs

Both numbers are positive: add their absolute values

\(|+7| + |+7| = 7 + 7 = 14\)

Step 3: Keep the common positive sign

Since both were positive, the result is positive: \(+14\)

Step 4: Write the final answer

\(+14\) or simply \(14\)

\( (+7) - (-7) = 14 \)
Final answer:

\( (+7) - (-7) = 14 \)

Applied rules:

• Subtraction rule: \(a - (-b) = a + b\)

• Same signs: Add absolute values, keep the common sign

• Pattern: \( (+a) - (-a) = +2a \)

Key Concepts: Laws, Methods, Rules, Definitions
\( a - b = a + (-b) \)
Fundamental Subtraction Rule
Key definitions:

Integer: A whole number that can be positive, negative, or zero (..., -3, -2, -1, 0, 1, 2, 3, ...)

Absolute value: The distance of a number from zero on the number line, denoted as \(|a|\), always non-negative

Positive integer: A number greater than zero, often written without a sign

Negative integer: A number less than zero, written with a minus sign

Opposite numbers: Two numbers that are the same distance from zero but in opposite directions

Subtraction rule: \(a - b = a + (-b)\) - Always convert subtraction to addition of the opposite

Complete subtraction methodology:
  1. Apply the subtraction rule: Change \(a - b\) to \(a + (-b)\)
  2. Identify signs: Determine if the resulting numbers have same or different signs
  3. Apply addition rules: Use the addition methods learned previously
  4. Same signs: Add absolute values and keep the common sign
  5. Different signs: Subtract absolute values and take the sign of the number with larger absolute value
  6. Verify: Check with number line or mental math
Tip 1: Remember "KCC": Keep the first number, Change the operation, Change the sign of the second number
Tip 2: Subtracting a negative is like adding a positive - the two negatives cancel out
Tip 3: Use a number line to visualize: start at first number, move in direction of the opposite of second number
Tip 4: When subtracting opposites, the result is twice the absolute value: \((+a) - (-a) = 2a\)
Common errors: Forgetting to change the sign of the number being subtracted, applying addition rules incorrectly, not converting to addition first.
Exam preparation: Master the KCC method, practice all four scenarios, verify answers with number lines.
Formulas to know by heart:

• Fundamental rule: \( a - b = a + (-b) \)

• Subtract positive: \( a - (+b) = a + (-b) \)

• Subtract negative: \( a - (-b) = a + b \)

• Opposite subtraction: \( (+a) - (-a) = +2a \) and \( (-a) - (+a) = -2a \)

• Identity: \( a - 0 = a \)

• Self-subtraction: \( a - a = 0 \)

Exercise with Visualization: Integer Subtraction Patterns
Exercise 6: Integer Subtraction on Number Line
Visualize these subtractions on a number line:
\( (+4) - (+7) \)
\( (-3) - (-5) \)
\( (+6) - (-2) \)

Analysis: The chart shows how integer subtraction works on the number line.

  • \( (+4) - (+7) = -3 \): Start at +4, move 7 left to -3
  • \( (-3) - (-5) = 2 \): Start at -3, move 5 right to 2
  • \( (+6) - (-2) = 8 \): Start at +6, move 2 right to 8

Questions & Answers

Question: I don't understand why subtracting a negative number gives a positive result. It seems backwards!

Answer: This is a very common source of confusion! Let me explain with a real-world analogy:

  • Think of temperature: If it's 5°C and you remove a cold spell (subtract a negative), it gets warmer
  • If you owe $5 (negative) and someone cancels a $3 debt (subtract -3), you're better off by $3

Mathematical explanation:

  • \(7 - (-3)\) means "start at 7, then go in the opposite direction of -3"
  • The opposite of -3 is +3, so you go 3 units to the right
  • That's the same as \(7 + 3 = 10\)

Remember the rule: Subtracting a negative is the same as adding a positive!

Question: What's the easiest way to remember the rules for subtracting integers?

Answer: Use the "KCC" method - it's very effective:

  • K - Keep the first number as it is
  • C - Change the subtraction sign to addition
  • C - Change the sign of the second number to its opposite

Examples:

  • \((+8) - (+3)\) → Keep +8, Change - to +, Change +3 to -3 → \((+8) + (-3) = 5\)
  • \((+6) - (-4)\) → Keep +6, Change - to +, Change -4 to +4 → \((+6) + (+4) = 10\)
  • \((-7) - (+2)\) → Keep -7, Change - to +, Change +2 to -2 → \((-7) + (-2) = -9\)

After using KCC, you'll have an addition problem that you already know how to solve!

Question: How do I know if my answer to an integer subtraction problem is reasonable?

Answer: Here are several ways to check if your answer is reasonable:

  1. Number line check: Draw a quick number line to visualize the movement
  2. Context check: Does the answer make sense in the problem context?
  3. Addition verification: Add your answer to the subtracted number - you should get the original number

Example verification: For \((-5) - (-9) = 4\)

  • Addition check: \(4 + (-9) = -5\) ✓ (back to original number)
  • Number line: Start at -5, move 9 units right (because subtracting -9 means going right), end at +4 ✓

The addition verification is especially reliable - if \(a - b = c\), then \(c + b = a\).