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Operations with Integers

Adding, subtracting, multiplying, and dividing integers

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Operations with Integers

What are Integers?

Integers are whole numbers and their opposites: ...,−3,−2,−1,0,1,2,3,......, -3, -2, -1, 0, 1, 2, 3, ...

Positive integers: 1,2,3,...1, 2, 3, ... Negative integers: −1,−2,−3,...-1, -2, -3, ... Zero: Neither positive nor negative

Adding Integers

Same signs: Add the absolute values, keep the sign

  • 5+3=85 + 3 = 8
  • −5+(−3)=−8-5 + (-3) = -8

Different signs: Subtract the absolute values, use the sign of the larger

  • 5+(−3)=25 + (-3) = 2
  • −5+3=−2-5 + 3 = -2

Subtracting Integers

Key rule: Subtracting is the same as adding the opposite

a−b=a+(−b)a - b = a + (-b)

Examples:

  • 7−3=7+(−3)=47 - 3 = 7 + (-3) = 4
  • 7−(−3)=7+3=107 - (-3) = 7 + 3 = 10
  • −7−3=−7+(−3)=−10-7 - 3 = -7 + (-3) = -10

Multiplying Integers

Sign rules:

  • Same signs → positive result
  • Different signs → negative result

Examples:

  • 5×3=155 \times 3 = 15 (both positive)
  • −5×(−3)=15-5 \times (-3) = 15 (both negative)
  • 5×(−3)=−155 \times (-3) = -15 (different signs)
  • −5×3=−15-5 \times 3 = -15 (different signs)

Dividing Integers

Same sign rules as multiplication:

  • Same signs → positive result
  • Different signs → negative result

Examples:

  • 12÷3=412 \div 3 = 4
  • −12÷(−3)=4-12 \div (-3) = 4
  • 12÷(−3)=−412 \div (-3) = -4
  • −12÷3=−4-12 \div 3 = -4

📚 Practice Problems

1Problem 1easy

❓ Question:

Calculate: −8+5-8 + 5

💡 Show Solution

Different signs: subtract and use sign of larger absolute value.

∣−8∣=8,∣5∣=5|-8| = 8, \quad |5| = 5

Since 8 > 5: 8−5=38 - 5 = 3

Use negative sign (from -8): −8+5=−3-8 + 5 = -3

Answer: −3-3

2Problem 2easy

❓ Question:

Calculate: −8+5-8 + 5

💡 Show Solution

Different signs: subtract and use sign of larger absolute value.

∣−8∣=8,∣5∣=5|-8| = 8, \quad |5| = 5

Since 8 > 5: 8−5=38 - 5 = 3

Use negative sign (from -8): −8+5=−3-8 + 5 = -3

Answer: −3-3

3Problem 3medium

❓ Question:

Calculate: −6−(−9)-6 - (-9)

💡 Show Solution

Change subtraction to adding the opposite: −6−(−9)=−6+9-6 - (-9) = -6 + 9

Different signs: subtract and use sign of larger. 9−6=39 - 6 = 3

Use positive sign (from 9): −6+9=3-6 + 9 = 3

Answer: 33

4Problem 4medium

❓ Question:

Calculate: −6−(−9)-6 - (-9)

💡 Show Solution

Change subtraction to adding the opposite: −6−(−9)=−6+9-6 - (-9) = -6 + 9

Different signs: subtract and use sign of larger. 9−6=39 - 6 = 3

Use positive sign (from 9): −6+9=3-6 + 9 = 3

Answer: 33

5Problem 5hard

❓ Question:

Calculate: (−4)×3×(−2)(-4) \times 3 \times (-2)

💡 Show Solution

Work left to right:

Step 1: (−4)×3=−12(-4) \times 3 = -12 (different signs → negative)

Step 2: −12×(−2)=24-12 \times (-2) = 24 (same signs → positive)

Answer: 2424

6Problem 6hard

❓ Question:

Calculate: (−4)×3×(−2)(-4) \times 3 \times (-2)

💡 Show Solution

Work left to right:

Step 1: (−4)×3=−12(-4) \times 3 = -12 (different signs → negative)

Step 2: −12×(−2)=24-12 \times (-2) = 24 (same signs → positive)

Answer: 2424

7Problem 7medium

❓ Question:

Calculate: (-48) ÷ 6

💡 Show Solution

Step 1: Identify the signs. Negative ÷ Positive = Negative

Step 2: Divide the absolute values. 48 ÷ 6 = 8

Step 3: Apply the negative sign. (-48) ÷ 6 = -8

Answer: -8

8Problem 8hard

❓ Question:

A scuba diver starts at the surface (0 feet), descends 40 feet, rises 15 feet, descends another 25 feet, then rises 30 feet. What is the diver's final depth?

💡 Show Solution

Step 1: Set up with integers. Surface = 0 Descend = negative (down) Rise = positive (up)

Step 2: Write the expression. 0 + (-40) + 15 + (-25) + 30

Step 3: Calculate step by step. 0 - 40 = -40 -40 + 15 = -25 -25 - 25 = -50 -50 + 30 = -20

Step 4: Interpret the result. -20 feet means 20 feet below surface

Answer: 20 feet below the surface (or -20 feet)

Explain using:

⚠️ Common Mistakes: Operations with Integers

Avoid these 3 frequent errors

🌍 Real-World Applications: Operations with Integers

See how this math is used in the real world

📝 Worked Example: Solving a Quadratic by Factoring

Problem:

Solve x2−5x+6=0x^2 - 5x + 6 = 0.

2Factor the quadratic
3Set each factor equal to zero
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Absolute Value
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📌 Related Topics in Integers and Rational Numbers

Absolute ValueAbsolute ValueComparing and Ordering Rational Numbers

❓ Frequently Asked Questions

What is Operations with Integers?▾
Adding, subtracting, multiplying, and dividing integers
How can I study Operations with Integers effectively?▾
Start by reading the study notes and working through the examples on this page. Then use the flashcards to test your recall. Practice with the 8 problems provided, checking solutions as you go. Regular review and active practice are key to retention.
Is this Operations with Integers study guide free?▾
Yes — all study notes, flashcards, and practice problems for Operations with Integers on Study Mondo are free to access. No account is needed.
What course covers Operations with Integers?▾
Operations with Integers is part of the Pre-Algebra course on Study Mondo, specifically in the Integers and Rational Numbers section. You can explore the full course for more related topics and practice resources.
Are there practice problems for Operations with Integers?▾
Yes, this page includes 8 practice problems with detailed solutions. Each problem includes a step-by-step explanation to help you understand the approach.