Integers Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Integers.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The set of whole numbers extended in both directions: positive whole numbers, their negatives, and zero.

Temperature can go above or below zero—integers include both directions.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Integers are the counting numbers, their negatives, and zero — direction and distance, no fractions.

Common stuck point: The procedure for integers is the easy part; the trap is thinking -3 is greater than -1 because 3 > 1. Asking "Is the value a whole amount that can be positive, negative, or zero (no fraction part)?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Is the value a whole amount that can be positive, negative, or zero (no fraction part)?

Common Mistakes to Watch For

Before you work through the examples, skim the mistake guide so you know which shortcuts and sign errors to avoid.

Negative Numbers Mistakes

Worked Examples

Example 1

easy
Evaluate (−8)+15+(−3)(-8) + 15 + (-3).

Answer

44

First step

1
Group the positive and negative terms: positives =15= 15, negatives =(−8)+(−3)=−11= (-8) + (-3) = -11.

Full solution

  1. 2
    Combine: 15+(−11)=15−11=415 + (-11) = 15 - 11 = 4.
  2. 3
    The result is 44.
When adding integers, group the positives and negatives separately, then find the difference. The sign of the result matches the group with the larger absolute value.

Example 2

medium
Evaluate (−6)×4÷(−3)(-6) \times 4 \div (-3).

Example 3

easy
Place −4-4, 00, 33, −1-1 on a number line and order least to greatest.

Example 4

medium
Order least to greatest: −7,∣−9∣,0,−∣−3∣,4-7, |-9|, 0, -|-3|, 4.

Example 5

hard
The product of two consecutive integers is 156156. Find them.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Evaluate (−12)−(−5)+3(-12) - (-5) + 3.

Example 2

easy
Evaluate (−18)÷3+7(-18) \div 3 + 7.

Example 3

easy
Compute −7+5-7 + 5.

Example 4

easy
Compute −3+(−4)-3 + (-4).

Example 5

easy
Compute 8−128 - 12.

Example 6

easy
Compute −9−(−3)-9 - (-3).

Example 7

easy
Compute 4×(−3)4 \times (-3).

Example 8

easy
Compute (−5)×(−6)(-5) \times (-6).

Example 9

easy
Compute (−20)÷4(-20) \div 4.

Example 10

easy
Compute ∣−15∣|-15|.

Example 11

medium
Compute −8+12−15+3-8 + 12 - 15 + 3.

Example 12

medium
Compute (−2)3(-2)^3.

Example 13

medium
Compute (−4)2−3×(−5)(-4)^2 - 3 \times (-5).

Example 14

medium
Temperature was −7°-7°C at 6 AM and rose by 15°15°. What is the new temperature?

Example 15

medium
Compute −(−(−7))-(-(-7)).

Example 16

medium
Compute −36−9\frac{-36}{-9}.

Example 17

medium
Order from least to greatest: −5,−2,3,−8,0-5, -2, 3, -8, 0.

Example 18

medium
Evaluate ∣−3∣+∣5∣−∣−2∣|-3| + |5| - |{-2}|.

Example 19

medium
Compute (−1)100(-1)^{100}.

Example 20

challenge
Find all integers nn such that −3<n<5-3 < n < 5.

Example 21

challenge
Find the smallest integer nn such that n2>100n^2 > 100.

Example 22

challenge
If a+b=−3a + b = -3 and ab=−10ab = -10, find aa and bb (integers).

Example 23

easy
Which is larger: −6-6 or −9-9?

Example 24

easy
List the integers between −3-3 and 22, exclusive.

Example 25

easy
Is −7.5-7.5 an integer?

Example 26

easy
What is the opposite of −13-13?

Example 27

easy
Is the sum of two integers always an integer?

Example 28

medium
Compute −13+27−8+4-13 + 27 - 8 + 4.

Example 29

medium
Compute (−3)4(-3)^4.

Example 30

medium
Find ∣−3∣−∣7∣+∣−2∣|-3| - |7| + |-2|.

Example 31

medium
A scuba diver is at −25-25 m. She rises 1010 m, then descends 77 m. Find her new depth.

Example 32

medium
Find (−2)5(-2)^5.

Example 33

medium
Is the set of integers closed under division? Give an example.

Example 34

medium
Find all integers nn with ∣n∣≤3|n| \le 3.

Example 35

hard
Find the smallest integer nn such that n2≥50n^2 \ge 50 with nn positive.

Example 36

hard
If aa and bb are integers with a>ba > b, is a2>b2a^2 > b^2 always true? Give an example.

Example 37

hard
How many integers nn satisfy −10≤2n≤8-10 \le 2n \le 8?

Example 38

hard
For integers x,yx, y with x+y=12x + y = 12 and x−y=2x - y = 2, find xx and yy.

Example 39

hard
Find all integer pairs (x,y)(x, y) with x2+y2=25x^2 + y^2 = 25.

Example 40

challenge
Sum of three consecutive integers is −93-93. Find them.

Example 41

challenge
Find the sum S=1−2+3−4+⋯+99−100S = 1 - 2 + 3 - 4 + \cdots + 99 - 100.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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