problem solving about integers

Integer Word Problems Worksheets

An integer is defined as a number that can be written without a fractional component. For example, 11, 8, 0, and −1908 are integers whereas √5, Π are not integers. The set of integers consists of zero, the positive natural numbers, and their additive inverses. Integers are closed under the operations of addition and multiplication . Integer word problems worksheets provide a variety of word problems associated with the use and properties of integers. 

Benefits of Integers Word Problems Worksheets

We use integers in our day-to-day life like measuring temperature, sea level, and speed limit. Translating verbal descriptions into expressions is an essential initial step in solving word problems. Deposits are normally represented by a positive sign and withdrawals are denoted by a negative sign. Negative numbers are used in weather forecasting to show the temperature of a region. Solving these integers word problems will help us relate the concept with practical applications.

Download Integers Word Problems Worksheet PDFs

These math worksheets should be practiced regularly and are free to download in PDF formats.

☛ Check Grade wise Integers Word Problems Worksheets

• 6th Grade Integers Worksheets
• Integers Worksheets for Grade 7
• 8th Grade Integers Worksheets

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How to Solve Integers and Their Properties

Last Updated: April 6, 2024

This article was reviewed by Joseph Meyer . Joseph Meyer is a High School Math Teacher based in Pittsburgh, Pennsylvania. He is an educator at City Charter High School, where he has been teaching for over 7 years. Joseph is also the founder of Sandbox Math, an online learning community dedicated to helping students succeed in Algebra. His site is set apart by its focus on fostering genuine comprehension through step-by-step understanding (instead of just getting the correct final answer), enabling learners to identify and overcome misunderstandings and confidently take on any test they face. He received his MA in Physics from Case Western Reserve University and his BA in Physics from Baldwin Wallace University. This article has been viewed 30,655 times.

An integer is a set of natural numbers, their negatives, and zero. However, some integers are natural numbers, including 1, 2, 3, and so on. Their negative values are, -1, -2, -3, and so on. So integers are the set of numbers including (…-3, -2, -1, 0, 1, 2, 3,…). An integer is never a fraction, decimal, or percentage, it can only be a whole number. To solve integers and use their properties, learn to use addition and subtraction properties and use multiplication properties.

Using Addition and Subtraction Properties

• a + b = c (where both a and b are positive numbers the sum c is also positive)
• For example: 2 + 2 = 4

• -a + -b = -c (where both a and b are negative, you get the absolute value of the numbers then you proceed to add, and use the negative sign for the sum)
• For example: -2+ (-2)=-4

• a + (-b) = c (when your terms are of different signs, determine the larger number's value, then get the absolute value of both terms and subtract the lesser value from the larger value. Use the sign of the larger number for the answer.)
• For example: 5 + (-1) = 4

• -a +b = c (get the absolute value of the numbers and again, proceed to subtract the lesser value from the larger value and assume the sign of the larger value)
• For example: -5 + 2 = -3

• An example of the additive identity is: a + 0 = a
• Mathematically, the additive identity looks like: 2 + 0 = 2 or 6 + 0 = 6

• The additive inverse is when a number is added to the negative equivalent of itself.
• For example: a + (-b) = 0, where b is equal to a
• Mathematically, the additive inverse looks like: 5 + -5 = 0

• For example: (5+3) +1 = 9 has the same sum as 5+ (3+1) = 9

Using Multiplication Properties

• When a and b are positive numbers and not equal to zero: +a * + b = +c
• When a and b are both negative numbers and not equal to zero: -a*-b = +c

• However, understand that any number multiplied by zero, equals zero.

• For example: a(b+c) = ab + ac
• Mathematically, this looks like: 5(2+3) = 5(2) + 5(3)
• Note that there is no inverse property for multiplication because the inverse of a whole number is a fraction, and fractions are not an element of integer.

Joseph Meyer

The distributive property helps you avoid repetitive calculations. You can use the distributive property to solve equations where you must multiply a number by a sum or difference. It simplifies calculations, enables expression manipulation (like factoring), and forms the basis for solving many equations.

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• Math Article
• Word Problems On Integers

Integers: Word Problems On Integers

An arithmetic operation is an elementary branch of mathematics. Arithmetical operations include addition, subtraction, multiplication and division. Arithmetic operations are applicable to different types of numbers including integers.

Integers are a special group of numbers that do not have a fractional or a decimal part. It includes positive numbers, negative numbers and zero.  Arithmetic operations on integers are similar to that of whole numbers. Since integers can be positive or negative numbers i.e. as these numbers are preceded either by a positive (+) or a negative sign (-), it makes them a little confusing concept. Therefore, they are different from whole numbers . Let us now see how various arithmetical operations can be performed on integers with the help of a few word problems. Solve the following word problems using various rules of operations of integers.

Word problems on integers Examples:

Example 1: Shyak has overdrawn his checking account by Rs.38.  The bank debited him Rs.20 for an overdraft fee.  Later, he deposited Rs.150.  What is his current balance?

Solution:  Given,

Total amount deposited= Rs. 150

Amount overdrew by Shyak= Rs. 38

Amount charged by bank= Rs. 20

⇒ Debit amount= -20

Total amount debited = (-38) + (-20) = -58

Current balance= Total deposit +Total Debit

Hence, the current balance is Rs. 92.

Example 2: Anna is a microbiology student. She was doing research on optimum temperature for the survival of different strains of bacteria. Studies showed that bacteria X need optimum temperature of -31˚C while bacteria Y need optimum temperature of -56˚C. What is the temperature difference?

Solution: Given,

Optimum temperature for bacteria X = -31˚C

Optimum temperature for bacteria Y= -56˚C

Temperature difference= Optimum temperature for bacteria X – Optimum temperature for bacteria Y

⇒ (-31) – (-56)

Hence, temperature difference is 25˚C.

Example 3: A submarine submerges at the rate of 5 m/min. If it descends from 20 m above the sea level, how long will it take to reach 250 m below sea level?

Initial position = 20 m    (above sea level)

Final position = 250 m    (below sea level)

Total depth it submerged = (250+20) = 270 m

Thus, the submarine travelled 270 m below sea level.

Time taken to submerge 1 meter = 1/5 minutes

Time taken to submerge 270 m = 270 (1/5) = 54 min

Hence, the submarine will reach 250 m below sea level in 54 minutes.

To solve more problems on the topic, download BYJU’S – The Learning App and watch interactive videos. Also, take free tests to practice for exams.

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Integers Worksheets

Welcome to the integers worksheets page at Math-Drills.com where you may have a negative experience, but in the world of integers, that's a good thing! This page includes Integers worksheets for comparing and ordering integers, adding, subtracting, multiplying and dividing integers and order of operations with integers.

If you've ever spent time in Canada in January, you've most likely experienced a negative integer first hand. Banks like you to keep negative balances in your accounts, so they can charge you loads of interest. Deep sea divers spend all sorts of time in negative integer territory. There are many reasons why a knowledge of integers is helpful even if you are not going to pursue an accounting or deep sea diving career. One hugely important reason is that there are many high school mathematics topics that will rely on a strong knowledge of integers and the rules associated with them.

We've included a few hundred integers worksheets on this page to help support your students in their pursuit of knowledge. You may also want to get one of those giant integer number lines to post if you are a teacher, or print off a few of our integer number lines. You can also project them on your whiteboard or make an overhead transparency. For homeschoolers or those with only one or a few students, the paper versions should do. The other thing that we highly recommend are integer chips a.k.a. two-color counters. Read more about them below.

Most Popular Integers Worksheets this Week

Integer Resources

Coordinate graph paper can be very useful when studying integers. Coordinate geometry is a practical application of integers and can give students practice with using integers while learning another related skill. Coordinate graph paper can be found on the Graph Paper page:

Coordinate Graph Paper

Integer number lines can be used for various math activities including operations with integers, counting, comparing, ordering, etc.

• Integer Number Lines Integers Number Lines from -10 to 10 Integers Number Lines from -15 to 15 Integers Number Lines from -20 to 20 Integers Number Lines from -25 to 25 OLD Integer Number Lines

Comparing and Ordering Integers

For students who are just starting with integers, it is very helpful if they can use an integer number line to compare integers and to see how the placement of integers works. They should quickly realize that negative numbers are counter-intuitive because they are probably quite used to larger absolute values meaning larger numbers. The reverse is the case, of course, with negative numbers. Students should be able to recognize easily that a positive number is always greater than a negative number and that between two negative integers, the one with the lesser absolute value is actually the greater number. Have students practice with these integers worksheets and follow up with the close proximity comparing integers worksheets.

• Comparing Integers Worksheets Comparing Positive and Negative Integers (-9 to +9) Comparing Positive and Negative Integers (-15 to +15) Comparing Positive and Negative Integers (-25 to +25) Comparing Positive and Negative Integers (-50 to +50) Comparing Positive and Negative Integers (-99 to +99) Comparing Negative Integers (-15 to -1)

By close proximity, we mean that the integers being compared differ very little in value. Depending on the range, we have allowed various differences between the two integers being compared. In the first set where the range is -9 to 9, the difference between the two numbers is always 1. With the largest range, a difference of up to 5 is allowed. These worksheets will help students further hone their ability to visualize and conceptualize the idea of negative numbers and will serve as a foundation for all the other worksheets on this page.

• Comparing Integers in Close Proximity Comparing Positive and Negative Integers (-9 to +9) in Close Proximity Comparing Positive and Negative Integers (-15 to +15) in Close Proximity Comparing Positive and Negative Integers (-25 to +25) in Close Proximity Comparing Positive and Negative Integers (-50 to +50) in Close Proximity Comparing Positive and Negative Integers (-99 to +99) in Close Proximity
• Ordering Integers Worksheets Ordering Integers on a Number Line Ordering Integers (range -9 to 9) Ordering Integers (range -20 to 20) Ordering Integers (range -50 to 50) Ordering Integers (range -99 to 99) Ordering Integers (range -999 to 999) Ordering Negative Integers (range -9 to -1) Ordering Negative Integers (range -99 to -10) Ordering Negative Integers (range -999 to -100)

Adding and Subtracting Integers

Two-color counters are fantastic manipulatives for teaching and learning about integer addition. Two-color counters are usually plastic chips that come with yellow on one side and red on the other side. They might be available in other colors, so you'll have to substitute your own colors in the following description.

Adding with two-color counters is actually quite easy. You model the first number with a pile of chips flipped to the correct side and you also model the second number with a pile of chips flipped to the correct side; then you mash them all together, take out the zeros (if any) and behold, you have your answer! Need further elaboration? Read on!

The correct side means using red to model negative numbers and yellow to model positive numbers. You would model —5 with five red chips and 7 with seven yellow chips. Mashing them together should be straight forward although, you'll want to caution your students to be less exuberant than usual, so none of the chips get flipped. Taking out the zeros means removing as many pairs of yellow and red chips as you can. You can do this because —1 and 1 when added together equals zero (this is called the zero principle). If you remove the zeros, you don't affect the answer. The benefit of removing the zeros, however, is that you always end up with only one color and as a consequence, the answer to the integer question. If you have no chips left at the end, the answer is zero!

• Adding Integers Worksheets with 75 Questions Per Page (Some Parentheses) Adding Integers from -9 to 9 (75 Questions) ✎ Adding Integers from -12 to 12 (75 Questions) ✎ Adding Integers from -15 to 15 (75 Questions) ✎ Adding Integers from -20 to 20 (75 Questions) ✎ Adding Integers from -25 to 25 (75 Questions) ✎ Adding Integers from -50 to 50 (75 Questions) ✎ Adding Integers from -99 to 99 (75 Questions) ✎
• Adding Integers Worksheets with 75 Questions Per Page (All Parentheses) Adding Integers from (-9) to (+9) All Parentheses (75 Questions) ✎ Adding Integers from (-12) to (+12) All Parentheses (75 Questions) ✎ Adding Integers from (-15) to (+15) All Parentheses (75 Questions) ✎ Adding Integers from (-20) to (+20) All Parentheses (75 Questions) ✎ Adding Integers from (-25) to (+25) All Parentheses (75 Questions) ✎ Adding Integers from (-50) to (+50) All Parentheses (75 Questions) ✎ Adding Integers from (-99) to (+99) All Parentheses (75 Questions) ✎
• Adding Integers Worksheets with 75 Questions Per Page (No Parentheses) Adding Integers from -9 to 9 No Parentheses (75 Questions) ✎ Adding Integers from -12 to 12 No Parentheses (75 Questions) ✎ Adding Integers from -15 to 15 No Parentheses (75 Questions) ✎ Adding Integers from -20 to 20 No Parentheses (75 Questions) ✎ Adding Integers from -25 to 25 No Parentheses (75 Questions) ✎ Adding Integers from -50 to 50 No Parentheses (75 Questions) ✎ Adding Integers from -99 to 99 No Parentheses (75 Questions) ✎
• Adding Integers Worksheets with 50 Questions Per Page (Some Parentheses) Adding Integers from -9 to 9 (50 Questions) ✎ Adding Integers from -12 to 12 (50 Questions) ✎ Adding Integers from -15 to 15 (50 Questions) ✎ Adding Integers from -20 to 20 (50 Questions) ✎ Adding Integers from -25 to 25 (50 Questions) ✎ Adding Integers from -50 to 50 (50 Questions) ✎ Adding Integers from -99 to 99 (50 Questions) ✎
• Adding Integers Worksheets with 50 Questions Per Page (All Parentheses) Adding Integers from (-9) to (+9) All Parentheses (50 Questions) ✎ Adding Integers from (-12) to (+12) All Parentheses (50 Questions) ✎ Adding Integers from (-15) to (+15) All Parentheses (50 Questions) ✎ Adding Integers from (-20) to (+20) All Parentheses (50 Questions) ✎ Adding Integers from (-25) to (+25) All Parentheses (50 Questions) ✎ Adding Integers from (-50) to (+50) All Parentheses (50 Questions) ✎ Adding Integers from (-99) to (+99) All Parentheses (50 Questions) ✎
• Adding Integers Worksheets with 50 Questions Per Page (No Parentheses) Adding Integers from -9 to 9 No Parentheses (50 Questions) ✎ Adding Integers from -12 to 12 No Parentheses (50 Questions) ✎ Adding Integers from -15 to 15 No Parentheses (50 Questions) ✎ Adding Integers from -20 to 20 No Parentheses (50 Questions) ✎ Adding Integers from -25 to 25 No Parentheses (50 Questions) ✎ Adding Integers from -50 to 50 No Parentheses (50 Questions) ✎ Adding Integers from -99 to 99 No Parentheses (50 Questions) ✎
• Adding Integers Worksheets with 25 Large Print Questions Per Page (Some Parentheses) Adding Integers from -9 to 9 (Large Print; 25 Questions) ✎ Adding Integers from -12 to 12 (Large Print; 25 Questions) ✎ Adding Integers from -15 to 15 (Large Print; 25 Questions) ✎ Adding Integers from -20 to 20 (Large Print; 25 Questions) ✎ Adding Integers from -25 to 25 (Large Print; 25 Questions) ✎ Adding Integers from -50 to 50 (Large Print; 25 Questions) ✎ Adding Integers from -99 to 99 (Large Print; 25 Questions) ✎
• Adding Integers Worksheets with 25 Large Print Questions Per Page (All Parentheses) Adding Integers from (-9) to (+9) All Parentheses (Large Print; 25 Questions) ✎ Adding Integers from (-12) to (+12) All Parentheses (Large Print; 25 Questions) ✎ Adding Integers from (-15) to (+15) All Parentheses (Large Print; 25 Questions) ✎ Adding Integers from (-20) to (+20) All Parentheses (Large Print; 25 Questions) ✎ Adding Integers from (-25) to (+25) All Parentheses (Large Print; 25 Questions) ✎ Adding Integers from (-50) to (+50) All Parentheses (Large Print; 25 Questions) ✎ Adding Integers from (-99) to (+99) All Parentheses (Large Print; 25 Questions) ✎
• Adding Integers Worksheets with 25 Large Print Questions Per Page (No Parentheses) Adding Integers from -9 to 9 No Parentheses (Large Print; 25 Questions) ✎ Adding Integers from -12 to 12 No Parentheses (Large Print; 25 Questions) ✎ Adding Integers from -15 to 15 No Parentheses (Large Print; 25 Questions) ✎ Adding Integers from -20 to 20 No Parentheses (Large Print; 25 Questions) ✎ Adding Integers from -25 to 25 No Parentheses (Large Print; 25 Questions) ✎ Adding Integers from -50 to 50 No Parentheses (Large Print; 25 Questions) ✎ Adding Integers from -99 to 99 No Parentheses (Large Print; 25 Questions) ✎
• Vertically Arranged Integer Addition Worksheets 3-Digit Integer Addition (Vertically Arranged) 3-Digit Positive Plus a Negative Integer Addition (Vertically Arranged) 3-Digit Negative Plus a Positive Integer Addition (Vertically Arranged) 3-Digit Negative Plus a Negative Integer Addition (Vertically Arranged)

Subtracting with integer chips is a little different. Integer subtraction can be thought of as removing. To subtract with integer chips, begin by modeling the first number (the minuend) with integer chips. Next, remove the chips that would represent the second number from your pile and you will have your answer. Unfortunately, that isn't all there is to it. This works beautifully if you have enough of the right color chip to remove, but often times you don't. For example, 5 - (-5), would require five yellow chips to start and would also require the removal of five red chips, but there aren't any red chips! Thank goodness, we have the zero principle. Adding or subtracting zero (a red chip and a yellow chip) has no effect on the original number, so we could add as many zeros as we wanted to the pile, and the number would still be the same. All that is needed then is to add as many zeros (pairs of red and yellow chips) as needed until there are enough of the correct color chip to remove. In our example 5 - (-5), you would add 5 zeros, so that you could remove five red chips. You would then be left with 10 yellow chips (or +10) which is the answer to the question.

• Subtracting Integers Worksheets with 75 Questions Per Page (Some Parentheses) Subtracting Integers from -9 to 9 (75 Questions) ✎ Subtracting Integers from -12 to 12 (75 Questions) ✎ Subtracting Integers from -15 to 15 (75 Questions) ✎ Subtracting Integers from -20 to 20 (75 Questions) ✎ Subtracting Integers from -25 to 25 (75 Questions) ✎ Subtracting Integers from -50 to 50 (75 Questions) ✎ Subtracting Integers from -99 to 99 (75 Questions) ✎
• Subtracting Integers Worksheets with 75 Questions Per Page (All Parentheses) Subtracting Integers from (-9) to (+9) All Parentheses (75 Questions) ✎ Subtracting Integers from (-12) to (+12) All Parentheses (75 Questions) ✎ Subtracting Integers from (-15) to (+15) All Parentheses (75 Questions) ✎ Subtracting Integers from (-20) to (+20) All Parentheses (75 Questions) ✎ Subtracting Integers from (-25) to (+25) All Parentheses (75 Questions) ✎ Subtracting Integers from (-50) to (+50) All Parentheses (75 Questions) ✎ Subtracting Integers from (-99) to (+99) All Parentheses (75 Questions) ✎
• Subtracting Integers Worksheets with 75 Questions Per Page (No Parentheses) Subtracting Integers from -9 to 9 No Parentheses (75 Questions) ✎ Subtracting Integers from -12 to 12 No Parentheses (75 Questions) ✎ Subtracting Integers from -15 to 15 No Parentheses (75 Questions) ✎ Subtracting Integers from -20 to 20 No Parentheses (75 Questions) ✎ Subtracting Integers from -25 to 25 No Parentheses (75 Questions) ✎ Subtracting Integers from -50 to 50 No Parentheses (75 Questions) ✎ Subtracting Integers from -99 to 99 No Parentheses (75 Questions) ✎
• Subtracting Integers Worksheets with 50 Questions Per Page (Some Parentheses) Subtracting Integers from -9 to 9 (50 Questions) ✎ Subtracting Integers from -12 to 12 (50 Questions) ✎ Subtracting Integers from -15 to 15 (50 Questions) ✎ Subtracting Integers from -20 to 20 (50 Questions) ✎ Subtracting Integers from -25 to 25 (50 Questions) ✎ Subtracting Integers from -50 to 50 (50 Questions) ✎ Subtracting Integers from -99 to 99 (50 Questions) ✎
• Subtracting Integers Worksheets with 50 Questions Per Page (All Parentheses) Subtracting Integers from (-9) to (+9) All Parentheses (50 Questions) ✎ Subtracting Integers from (-12) to (+12) All Parentheses (50 Questions) ✎ Subtracting Integers from (-15) to (+15) All Parentheses (50 Questions) ✎ Subtracting Integers from (-20) to (+20) All Parentheses (50 Questions) ✎ Subtracting Integers from (-25) to (+25) All Parentheses (50 Questions) ✎ Subtracting Integers from (-50) to (+50) All Parentheses (50 Questions) ✎ Subtracting Integers from (-99) to (+99) All Parentheses (50 Questions) ✎
• Subtracting Integers Worksheets with 50 Questions Per Page (No Parentheses) Subtracting Integers from -9 to 9 No Parentheses (50 Questions) ✎ Subtracting Integers from -12 to 12 No Parentheses (50 Questions) ✎ Subtracting Integers from -15 to 15 No Parentheses (50 Questions) ✎ Subtracting Integers from -20 to 20 No Parentheses (50 Questions) ✎ Subtracting Integers from -25 to 25 No Parentheses (50 Questions) ✎ Subtracting Integers from -50 to 50 No Parentheses (50 Questions) ✎ Subtracting Integers from -99 to 99 No Parentheses (50 Questions) ✎
• Subtracting Integers Worksheets with 25 Large Print Questions Per Page (Some Parentheses) Subtracting Integers from -9 to 9 (Large Print; 25 Questions) ✎ Subtracting Integers from -12 to 12 (Large Print; 25 Questions) ✎ Subtracting Integers from -15 to 15 (Large Print; 25 Questions) ✎ Subtracting Integers from -20 to 20 (Large Print; 25 Questions) ✎ Subtracting Integers from -25 to 25 (Large Print; 25 Questions) ✎ Subtracting Integers from -50 to 50 (Large Print; 25 Questions) ✎ Subtracting Integers from -99 to 99 (Large Print; 25 Questions) ✎
• Subtracting Integers Worksheets with 25 Large Print Questions Per Page (All Parentheses) Subtracting Integers from (-9) to (+9) All Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-12) to (+12) All Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-15) to (+15) All Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-20) to (+20) All Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-25) to (+25) All Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-50) to (+50) All Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-99) to (+99) All Parentheses (Large Print; 25 Questions) ✎
• Subtracting Integers Worksheets with 25 Large Print Questions Per Page (No Parentheses) Subtracting Integers from (-9) to 9 No Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-12) to 12 No Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-15) to 15 No Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-20) to 20 No Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-25) to 25 No Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-50) to 50 No Parentheses (Large Print; 25 Questions) ✎ Subtracting Integers from (-99) to 99 No Parentheses (Large Print; 25 Questions) ✎
• Vertically Arranged Integer Subtraction Worksheets 3-Digit Integer Subtraction (Vertically Arranged) 3-Digit Positive Minus a Positive Integer Subtraction (Vertically Arranged) 3-Digit Positive Minus a Negative Integer Subtraction (Vertically Arranged) 3-Digit Negative Minus a Positive Integer Subtraction (Vertically Arranged) 3-Digit Negative Minus a Negative Integer Subtraction (Vertically Arranged)

The worksheets in this section include addition and subtraction on the same page. Students will have to pay close attention to the signs and apply their knowledge of integer addition and subtraction to each question. The use of counters or number lines could be helpful to some students.

• Adding and Subtracting Integers Worksheets with 75 Questions Per Page (Some Parentheses) Adding & Subtracting Integers from -9 to 9 (75 Questions) ✎ Adding & Subtracting Integers from -10 to 10 (75 Questions) ✎ Adding & Subtracting Integers from -12 to 12 (75 Questions) ✎ Adding & Subtracting Integers from -15 to 15 (75 Questions) ✎ Adding & Subtracting Integers from -20 to 20 (75 Questions) ✎ Adding & Subtracting Integers from -25 to 25 (75 Questions) ✎ Adding & Subtracting Integers from -50 to 50 (75 Questions) ✎ Adding & Subtracting Integers from -99 to 99 (75 Questions) ✎
• Adding and Subtracting Integers Worksheets with 75 Questions Per Page (All Parentheses) Adding & Subtracting Integers from (-5) to (+5) All Parentheses (75 Questions) ✎ Adding & Subtracting Integers from (-9) to (+9) All Parentheses (75 Questions) ✎ Adding & Subtracting Integers from (-12) to (+12) All Parentheses (75 Questions) ✎ Adding & Subtracting Integers from (-15) to (+15) All Parentheses (75 Questions) ✎ Adding & Subtracting Integers from (-20) to (+20) All Parentheses (75 Questions) ✎ Adding & Subtracting Integers from (-25) to (+25) All Parentheses (75 Questions) ✎ Adding & Subtracting Integers from (-50) to (+50) All Parentheses (75 Questions) ✎ Adding & Subtracting Integers from (-99) to (+99) All Parentheses (75 Questions) ✎
• Adding and Subtracting Integers Worksheets with 75 Questions Per Page (No Parentheses) Adding & Subtracting Integers from -9 to 9 No Parentheses (75 Questions) ✎ Adding & Subtracting Integers from -12 to 12 No Parentheses (75 Questions) ✎ Adding & Subtracting Integers from -15 to 15 No Parentheses (75 Questions) ✎ Adding & Subtracting Integers from -20 to 20 No Parentheses (75 Questions) ✎ Adding & Subtracting Integers from -25 to 25 No Parentheses (75 Questions) ✎ Adding & Subtracting Integers from -50 to 50 No Parentheses (75 Questions) ✎ Adding & Subtracting Integers from -99 to 99 No Parentheses (75 Questions) ✎
• Adding and Subtracting Integers Worksheets with 50 Questions Per Page (Some Parentheses) Adding & Subtracting Integers from -9 to 9 (50 Questions) ✎ Adding & Subtracting Integers from -12 to 12 (50 Questions) ✎ Adding & Subtracting Integers from -15 to 15 (50 Questions) ✎ Adding & Subtracting Integers from -20 to 20 (50 Questions) ✎ Adding & Subtracting Integers from -25 to 25 (50 Questions) ✎ Adding & Subtracting Integers from -50 to 50 (50 Questions) ✎ Adding & Subtracting Integers from -99 to 99 (50 Questions) ✎
• Adding and Subtracting Integers Worksheets with 50 Questions Per Page (All Parentheses) Adding & Subtracting Integers from (-9) to (+9) All Parentheses (50 Questions) ✎ Adding & Subtracting Integers from (-12) to (+12) All Parentheses (50 Questions) ✎ Adding & Subtracting Integers from (-15) to (+15) All Parentheses (50 Questions) ✎ Adding & Subtracting Integers from (-20) to (+20) All Parentheses (50 Questions) ✎ Adding & Subtracting Integers from (-25) to (+25) All Parentheses (50 Questions) ✎ Adding & Subtracting Integers from (-50) to (+50) All Parentheses (50 Questions) ✎ Adding & Subtracting Integers from (-99) to (+99) All Parentheses (50 Questions) ✎
• Adding and Subtracting Integers Worksheets with 50 Questions Per Page (No Parentheses) Adding & Subtracting Integers from -9 to 9 No Parentheses (50 Questions) ✎ Adding & Subtracting Integers from -12 to 12 No Parentheses (50 Questions) ✎ Adding & Subtracting Integers from -15 to 15 No Parentheses (50 Questions) ✎ Adding & Subtracting Integers from -20 to 20 No Parentheses (50 Questions) ✎ Adding & Subtracting Integers from -25 to 25 No Parentheses (50 Questions) ✎ Adding & Subtracting Integers from -50 to 50 No Parentheses (50 Questions) ✎ Adding & Subtracting Integers from -99 to 99 No Parentheses (50 Questions) ✎
• Adding and Subtracting Integers Worksheets with 25 Large Print Questions Per Page (Some Parentheses) Adding & Subtracting Integers from -9 to 9 (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from -12 to 12 (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from -15 to 15 (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from -20 to 20 (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from -25 to 25 (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from -50 to 50 (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from -99 to 99 (Large Print; 25 Questions) ✎
• Adding and Subtracting Integers Worksheets with 25 Large Print Questions Per Page (All Parentheses) Adding & Subtracting Integers from (-9) to (+9) All Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-12) to (+12) All Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-15) to (+15) All Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-20) to (+20) All Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-25) to (+25) All Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-50) to (+50) All Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-99) to (+99) All Parentheses (Large Print; 25 Questions) ✎
• Adding and Subtracting Integers Worksheets with 25 Large Print Questions Per Page (No Parentheses) Adding & Subtracting Integers from (-9) to 9 No Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-12) to 12 No Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-15) to 15 No Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-20) to 20 No Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-25) to 25 No Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-50) to 50 No Parentheses (Large Print; 25 Questions) ✎ Adding & Subtracting Integers from (-99) to 99 No Parentheses (Large Print; 25 Questions) ✎

These worksheets include groups of questions that all result in positive or negative sums or differences. They can be used to help students see more clearly how certain integer questions end up with positive and negative results. In the case of addition of negative and positive integers, some people suggest looking for the "heavier" value to determine whether the sum will be positive of negative. More technically, it would be the integer with the greater absolute value. For example, in the question (−2) + 5, the absolute value of the positive integer is greater, so the sum will be positive.

In subtraction questions, the focus is on the subtrahend (the value being subtracted). In positive minus positive questions, if the subtrahend is greater than the minuend, the answer will be negative. In negative minus negative questions, if the subtrahend has a greater absolute value, the answer will be positive. Vice-versa for both situations. Alternatively, students can always convert subtraction questions to addition questions by changing the signs (e.g. (−5) − (−7) is the same as (−5) + 7; 3 − 5 is the same as 3 + (−5)).

• Scaffolded Integer Addition and Subtraction Positive Plus Negative Integer Addition (Scaffolded) ✎ Negative Plus Positive Integer Addition (Scaffolded) ✎ Mixed Integer Addition (Scaffolded) ✎ Positive Minus Positive Integer Subtraction (Scaffolded) ✎ Negative Minus Negative Integer Subtraction (Scaffolded) ✎

Multiplying and Dividing Integers

Multiplying integers is very similar to multiplication facts except students need to learn the rules for the negative and positive signs. In short, they are:

In words, multiplying two positives or two negatives together results in a positive product, and multiplying a negative and a positive in either order results in a negative product. So, -8 × 8, 8 × (-8), -8 × (-8) and 8 × 8 all result in an absolute value of 64, but in two cases, the answer is positive (64) and in two cases the answer is negative (-64).

Should you wish to develop some "real-world" examples of integer multiplication, it might be a stretch due to the abstract nature of negative numbers. Sure, you could come up with some scenario about owing a debt and removing the debt in previous months, but this may only result in confusion. For now students can learn the rules of multiplying integers and worry about the analogies later!

• Multiplying Integers with 100 Questions Per Page Multiplying Mixed Integers from -9 to 9 (100 Questions) ✎ Multiplying Positive by Negative Integers from -9 to 9 (100 Questions) ✎ Multiplying Negative by Positive Integers from -9 to 9 (100 Questions) ✎ Multiplying Negative by Negative Integers from -9 to 9 (100 Questions) ✎ Multiplying Mixed Integers from -12 to 12 (100 Questions) ✎ Multiplying Positive by Negative Integers from -12 to 12 (100 Questions) ✎ Multiplying Negative by Positive Integers from -12 to 12 (100 Questions) ✎ Multiplying Negative by Negative Integers from -12 to 12 (100 Questions) ✎ Multiplying Mixed Integers from -20 to 20 (100 Questions) ✎ Multiplying Mixed Integers from -50 to 50 (100 Questions) ✎
• Multiplying Integers with 50 Questions Per Page Multiplying Mixed Integers from -9 to 9 (50 Questions) ✎ Multiplying Positive by Negative Integers from -9 to 9 (50 Questions) ✎ Multiplying Negative by Positive Integers from -9 to 9 (50 Questions) ✎ Multiplying Negative by Negative Integers from -9 to 9 (50 Questions) ✎ Multiplying Mixed Integers from -12 to 12 (50 Questions) ✎ Multiplying Positive by Negative Integers from -12 to 12 (50 Questions) ✎ Multiplying Negative by Positive Integers from -12 to 12 (50 Questions) ✎ Multiplying Negative by Negative Integers from -12 to 12 (50 Questions) ✎
• Multiplying Integers with 25 Large Print Questions Per Page Multiplying Mixed Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying Positive by Negative Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying Negative by Positive Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying Negative by Negative Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying Mixed Integers from -12 to 12 (25 Questions; Large Print) ✎ Multiplying Positive by Negative Integers from -12 to 12 (25 Questions; Large Print) ✎ Multiplying Negative by Positive Integers from -12 to 12 (25 Questions; Large Print) ✎ Multiplying Negative by Negative Integers from -12 to 12 (25 Questions; Large Print) ✎

Luckily (for your students), the rules of dividing integers are the same as the rules for multiplying:

Dividing a positive by a positive integer or a negative by a negative integer will result in a positive integer. Dividing a negative by a positive integer or a positive by a negative integer will result in a negative integer. A good grasp of division facts and a knowledge of the rules for multiplying and dividing integers will go a long way in helping your students master integer division. Use the worksheets in this section to guide students along.

• Dividing Integers with 100 Questions Per Page Dividing Mixed Integers from -9 to 9 (100 Questions) ✎ Dividing Positive by Negative Integers from -9 to 9 (100 Questions) ✎ Dividing Negative by Positive Integers from -9 to 9 (100 Questions) ✎ Dividing Negative by Negative Integers from -9 to 9 (100 Questions) ✎ Dividing Mixed Integers from -12 to 12 (100 Questions) ✎ Dividing Positive by Negative Integers from -12 to 12 (100 Questions) ✎ Dividing Negative by Positive Integers from -12 to 12 (100 Questions) ✎ Dividing Negative by Negative Integers from -12 to 12 (100 Questions) ✎
• Dividing Integers with 50 Questions Per Page Dividing Mixed Integers from -9 to 9 (50 Questions) ✎ Dividing Positive by Negative Integers from -9 to 9 (50 Questions) ✎ Dividing Negative by Positive Integers from -9 to 9 (50 Questions) ✎ Dividing Negative by Negative Integers from -9 to 9 (50 Questions) ✎ Dividing Mixed Integers from -12 to 12 (50 Questions) ✎ Dividing Positive by Negative Integers from -12 to 12 (50 Questions) ✎ Dividing Negative by Positive Integers from -12 to 12 (50 Questions) ✎ Dividing Negative by Negative Integers from -12 to 12 (50 Questions) ✎
• Dividing Integers with 25 Large Print Questions Per Page Dividing Mixed Integers from -9 to 9 (25 Questions; Large Print) ✎ Dividing Positive by Negative Integers from -9 to 9 (25 Questions; Large Print) ✎ Dividing Negative by Positive Integers from -9 to 9 (25 Questions; Large Print) ✎ Dividing Negative by Negative Integers from -9 to 9 (25 Questions; Large Print) ✎ Dividing Mixed Integers from -12 to 12 (25 Questions; Large Print) ✎ Dividing Positive by Negative Integers from -12 to 12 (25 Questions; Large Print) ✎ Dividing Negative by Positive Integers from -12 to 12 (25 Questions; Large Print) ✎ Dividing Negative by Negative Integers from -12 to 12 (25 Questions; Large Print) ✎

This section includes worksheets with both multiplying and dividing integers on the same page. As long as students know their facts and the integer rules for multiplying and dividing, their sole worry will be to pay attention to the operation signs.

• Multiplying and Dividing Integers with 100 Questions Per Page Multiplying and Dividing Mixed Integers from -9 to 9 (100 Questions) ✎ Multiplying and Dividing Positive and Negative Integers from -9 to 9 (100 Questions) ✎ Multiplying and Dividing Negative and Positive Integers from -9 to 9 (100 Questions) ✎ Multiplying and Dividing Negative and Negative Integers from -9 to 9 (100 Questions) ✎ Multiplying and Dividing Mixed Integers from -12 to 12 (100 Questions) ✎ Multiplying and Dividing Positive and Negative Integers from -12 to 12 (100 Questions) ✎ Multiplying and Dividing Negative and Positive Integers from -12 to 12 (100 Questions) ✎ Multiplying and Dividing Negative and Negative Integers from -12 to 12 (100 Questions) ✎
• Multiplying and Dividing Integers with 75 Questions Per Page Multiplying and Dividing Mixed Integers from -9 to 9 (75 Questions) ✎ Multiplying and Dividing Positive and Negative Integers from -9 to 9 (75 Questions) ✎ Multiplying and Dividing Negative and Positive Integers from -9 to 9 (75 Questions) ✎ Multiplying and Dividing Negative and Negative Integers from -9 to 9 (75 Questions) ✎ Multiplying and Dividing Mixed Integers from -12 to 12 (75 Questions) ✎ Multiplying and Dividing Positive and Negative Integers from -12 to 12 (75 Questions) ✎ Multiplying and Dividing Negative and Positive Integers from -12 to 12 (75 Questions) ✎ Multiplying and Dividing Negative and Negative Integers from -12 to 12 (75 Questions) ✎
• Multiplying and Dividing Integers with 50 Questions Per Page Multiplying and Dividing Mixed Integers from -9 to 9 (50 Questions) ✎ Multiplying and Dividing Positive and Negative Integers from -9 to 9 (50 Questions) ✎ Multiplying and Dividing Negative and Positive Integers from -9 to 9 (50 Questions) ✎ Multiplying and Dividing Negative and Negative Integers from -9 to 9 (50 Questions) ✎ Multiplying and Dividing Mixed Integers from -12 to 12 (50 Questions) ✎ Multiplying and Dividing Positive and Negative Integers from -12 to 12 (50 Questions) ✎ Multiplying and Dividing Negative and Positive Integers from -12 to 12 (50 Questions) ✎ Multiplying and Dividing Negative and Negative Integers from -12 to 12 (50 Questions) ✎
• Multiplying and Dividing Integers with 25 Large Print Questions Per Page Multiplying and Dividing Mixed Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying and Dividing Positive and Negative Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying and Dividing Negative and Positive Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying and Dividing Negative and Negative Integers from -9 to 9 (25 Questions; Large Print) ✎ Multiplying and Dividing Mixed Integers from -12 to 12 (25 Questions; Large Print) ✎ Multiplying and Dividing Positive and Negative Integers from -12 to 12 (25 Questions; Large Print) ✎ Multiplying and Dividing Negative and Positive Integers from -12 to 12 (25 Questions; Large Print) ✎ Multiplying and Dividing Negative and Negative Integers from -12 to 12 (25 Questions; Large Print) ✎

All Operations with Integers

In this section, the integers math worksheets include all of the operations. Students will need to pay attention to the operations and the signs and use mental math or another strategy to arrive at the correct answers. It should go without saying that students need to know their basic addition, subtraction, multiplication and division facts and rules regarding operations with integers before they should complete any of these worksheets independently. Of course, the worksheets can be used as a source of questions for lessons, tests or other learning activities.

• All Operations with Integers with 50 Questions Per Page (Some Parentheses) All operations with integers from -9 to 9 (50 Questions) ✎ All operations with integers from -12 to 12 (50 Questions) ✎ All operations with integers from -15 to 15 (50 Questions) ✎ All operations with integers from -20 to 20 (50 Questions) ✎ All operations with integers from -25 to 25 (50 Questions) ✎ All operations with integers from -50 to 50 (50 Questions) ✎ All operations with integers from -99 to 99 (50 Questions) ✎
• All Operations with Integers with 50 Questions Per Page (All Parentheses) All operations with integers from (-9) to (+9) All Parentheses (50 Questions) ✎ All operations with integers from (-12) to (+12) All Parentheses (50 Questions) ✎ All operations with integers from (-15) to (+15) All Parentheses (50 Questions) ✎ All operations with integers from (-20) to (+20) All Parentheses (50 Questions) ✎ All operations with integers from (-25) to (+25) All Parentheses (50 Questions) ✎ All operations with integers from (-50) to (+50) All Parentheses (50 Questions) ✎ All operations with integers from (-99) to (+99) All Parentheses (50 Questions) ✎
• All Operations with Integers with 50 Questions Per Page (No Parentheses) All operations with integers from -9 to 9 No Parentheses (50 Questions) ✎ All operations with integers from -12 to 12 No Parentheses (50 Questions) ✎ All operations with integers from -15 to 15 No Parentheses (50 Questions) ✎ All operations with integers from -20 to 20 No Parentheses (50 Questions) ✎ All operations with integers from -25 to 25 No Parentheses (50 Questions) ✎ All operations with integers from -50 to 50 No Parentheses (50 Questions) ✎ All operations with integers from -99 to 99 No Parentheses (50 Questions) ✎

Order of operations with integers can be found on the Order of Operations page:

Order of Operations with Integers

Copyright © 2005-2024 Math-Drills.com You may use the math worksheets on this website according to our Terms of Use to help students learn math.

Algebra Word Problems: Integers

Things to watch out for: be careful when translating the sentences into equations.

For example: “John has 5 fewer sweets than twice the number that Alice has”         is translated as j = 2 a –5         and not j = 5 – 2 a

Although, the number 5 is mentioned first in the sentence that does not mean that it would come first in the equation. Read the sentence carefully. In these lessons, we will cover some examples of integer word problems with two unknowns. Refer to the following related topics for other types of integer word problems.

Integer Problems With Two Unknowns

A team won 3 times as many matches as it lost. If it won 15 matches, how many games did it lose?

Step 1: Sentence: A team won 3 times as many matches as it lost.

Assign variables :

Sentence: It won 15 matches 3x = 15

Step 2: Solve the equation

Initially, there were the same number of blue marbles and red marbles in a bag. John took out 5 blue marbles and then there were twice as many red marbles as blue marbles in the bag. How many red marbles are there in the bag?

Step 1: Assign variables :

Let x = red marbles

Sentence: Initially, blue marbles = red marbles = x , then John took out 5 blue marbles. x – 5 = blue marbles

Sentence: twice as many red marbles as blue marbles in the bag x = 2( x –5)

x = 2( x –5)

Remove the brackets x = 2 x – 10

Isolate variable x x = 10

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1.3: Integers

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By the end of this section, you will be able to:

• Simplify expressions with absolute value
• Add and subtract integers
• Multiply and divide integers
• Simplify expressions with integers
• Evaluate variable expressions with integers
• Translate phrases to expressions with integers
• Use integers in applications

A more thorough introduction to the topics covered in this section can be found in the Elementary Algebra chapter, Foundations.

Simplify Expressions with Absolute Value

A negative number is a number less than 0. The negative numbers are to the left of zero on the number line ( Figure \(\PageIndex{1}\)).

You may have noticed that, on the number line, the negative numbers are a mirror image of the positive numbers, with zero in the middle. Because the numbers \(2\) and \(−2\) are the same distance from zero, each one is called the opposite of the other. The opposite of \(2\) is \(−2\), and the opposite of \(−2\) is \(2\).

The opposite of a number is the number that is the same distance from zero on the number line but on the opposite side of zero.

Figure \(\PageIndex{2}\) illustrates the definition.

OPPOSITE NOTATION

\[\begin{align} & -a \text{ means the opposite of the number }a \\ & \text{The notation} -a \text{ is read as “the opposite of }a \text{.”} \end{align} \]

We saw that numbers such as 3 and −3 are opposites because they are the same distance from 0 on the number line. They are both three units from 0. The distance between 0 and any number on the number line is called the absolute value of that number.

Definition: ABSOLUTE VALUE

The absolute value of a number is its distance from 0 on the number line.

The absolute value of a number \(n\) is written as \(|n|\) and \(|n|≥0\) for all numbers.

Absolute values are always greater than or equal to zero.

For example,

\[\begin{align} & -5 \text{ is } 5 \text{ units away from 0, so } |-5|=5. \\ & 5 \text{ is }5\text{ units away from 0, so }|5|=5. \end{align}\]

Figure \(\PageIndex{3}\) illustrates this idea.

The absolute value of a number is never negative because distance cannot be negative. The only number with absolute value equal to zero is the number zero itself because the distance from 0 to 0 on the number line is zero units.

In the next example, we’ll order expressions with absolute values.

EXAMPLE \(\PageIndex{1}\)

Fill in \(<,\,>,\) or \(=\) for each of the following pairs of numbers:

• \(\mathrm{|−5|}\_\_\mathrm{−|−5|}\_\_\mathrm{−|5|}\)
• \(\text{8__−|−8|}\)
• \(\text{−9__−|−9|}\)
• (\text{−(−16)__|−16|}\).

\(\begin{array}{lrcc} { \text{ } \\ \text{Simplify.} \\ \text{Order.} \\ \text{ } } & {|−5| \\ 5 \\ 5 \\ |−5|} & {\_\_ \\ \_\_ \\ > \\ >} & {−|−5| \\ −5 \\ −5 \\ −|−5|} \end{array}\)

\(\begin{array}{llcc} { \text{ } \\ \text{Simplify.} \\ \text{Order.} \\ \text{ } } & {8 \\ 8 \\ 8 \\ 8} & {\_\_ \\ \_\_ \\ > \\ >} & {−|−8| \\ −8 \\ −8 \\ −|−8|} \end{array}\)

\(\begin{array}{lrcc} { \text{ } \\ \text{Simplify.} \\ \text{Order.} \\ \text{ } } & {−9 \\ −9 \\ −9 \\ −9} & {\_\_ \\ \_\_ \\ = \\ =} & {−|−9| \\ −9 \\ −9 \\ −|−9|} \end{array}\)

\(\begin{array}{lrcc} { \text{ } \\ \text{Simplify.} \\ \text{Order.} \\ \text{ } } & {−(−16) \\ 16 \\ 16 \\ −(−16)} & {\_\_ \\ \_\_ \\ = \\ =} & {−|−16| \\ 16 \\ 16 \\ |−16|} \end{array}\)

EXAMPLE \(\PageIndex{2}\)

ⓐ \(−9 \_\_−|−9|\) ⓑ \(2 \_\_−|−2|\) ⓒ \(−8 \_\_|−8|\) ⓓ \(−(−9) \_\_|−9|.\)

ⓐ \(>\) ⓑ \(>\) ⓒ \(<\)

ⓓ \(=\)

EXAMPLE \(\PageIndex{3}\)

Fill in \(<,>,\) or \(=\) for each of the following pairs of numbers:

• \(7 \_\_ −|−7|\)
• \(−(−10) \_ \_|−10|\)
• \(|−4| \_\_ −|−4|\)
• \(−1 \_\_ |−1|.\)

ⓐ \(>\) ⓑ \(=\) ⓒ \(>\)

ⓓ \(<\)

We now add absolute value bars to our list of grouping symbols. When we use the order of operations, first we simplify inside the absolute value bars as much as possible, then we take the absolute value of the resulting number.

GROUPING SYMBOLS

• \[\begin{array}{lclc} \text{Parentheses} & () & \text{Braces} & \{ \} \\ \text{Brackets} & [] & \text{Absolute value} & ||\end{array}\]

In the next example, we simplify the expressions inside absolute value bars first just like we do with parentheses.

EXAMPLE \(\PageIndex{4}\)

Simplify: \(\mathrm{24−|19−3(6−2)|}\).

\(\begin{array}{lc} \text{} & 24−|19−3(6−2)| \\ \text{Work inside parentheses first:} & \text{} \\ \text{subtract 2 from 6.} & 24−|19−3(4)| \\ \text{Multiply 3(4).} & 24−|19−12| \\ \text{Subtract inside the absolute value bars.} & 24−|7| \\ \text{Take the absolute value.} & 24−7 \\ \text{Subtract.} & 17 \end{array}\)

EXAMPLE \(\PageIndex{5}\)

Simplify: \(19−|11−4(3−1)|\).

EXAMPLE \(\PageIndex{6}\)

Simplify: \(9−|8−4(7−5)|\).

Add and Subtract Integers

So far in our examples, we have only used the counting numbers and the whole numbers.

\[\begin{array}{ll} \text{Counting numbers} & 1,2,3… \\ \text{Whole numbers} 0,1,2,3…. \end{array}\]

Our work with opposites gives us a way to define the integers . The whole numbers and their opposites are called the integers. The integers are the numbers \(…−3,−2,−1,0,1,2,3…\)

Definition: INTEGERS

The whole numbers and their opposites are called the integers .

The integers are the numbers

\[…-3,-2,-1,0,1,2,3…,\]

Most students are comfortable with the addition and subtraction facts for positive numbers. But doing addition or subtraction with both positive and negative numbers may be more challenging.

We will use two color counters to model addition and subtraction of negatives so that you can visualize the procedures instead of memorizing the rules.

We let one color (blue) represent positive. The other color (red) will represent the negatives.

If we have one positive counter and one negative counter, the value of the pair is zero. They form a neutral pair. The value of this neutral pair is zero.

We will use the counters to show how to add:

\[5+3 \; \; \; \; \; \; −5+(−3) \; \; \; \; \; \; −5+3 \; \; \; \; \; \; \; 5+(−3)\]

The first example, \(5+3,\) adds 5 positives and 3 positives—both positives.

The second example, \(−5+(−3),\) adds 5 negatives and 3 negatives—both negatives.

When the signs are the same, the counters are all the same color, and so we add them. In each case we get 8—either 8 positives or 8 negatives.

So what happens when the signs are different? Let’s add \(−5+3\) and \(5+(−3)\).

When we use counters to model addition of positive and negative integers, it is easy to see whether there are more positive or more negative counters. So we know whether the sum will be positive or negative.

EXAMPLE \(\PageIndex{7}\)

Add: ⓐ \(−1+(−4)\) ⓑ \(−1+5\) ⓒ \(1+(−5)\).

EXAMPLE \(\PageIndex{8}\)

Add: ⓐ \(−2+(−4)\) ⓑ \(−2+4\) ⓒ \(2+(−4)\).

ⓐ \(−6\) ⓑ \(2\) ⓒ \(−2\)

EXAMPLE \(\PageIndex{9}\)

Add: ⓐ \(−2+(−5)\) ⓑ \(−2+5\) ⓒ \(2+(−5)\).

ⓐ \(−7\) ⓑ \(3\) ⓒ \(−3\)

We will continue to use counters to model the subtraction. Perhaps when you were younger, you read \(“5−3”\) as “5 take away 3.” When you use counters, you can think of subtraction the same way!

We will use the counters to show to subtract:

\[5−3 \; \; \; \; \; \; −5−(−3) \; \; \; \; \; \; −5−3 \; \; \; \; \; \; 5−(−3) \]

The first example, \(5−3\), we subtract 3 positives from 5 positives and end up with 2 positives.

In the second example, \(−5−(−3),\) we subtract 3 negatives from 5 negatives and end up with 2 negatives.

Each example used counters of only one color, and the “take away” model of subtraction was easy to apply.

What happens when we have to subtract one positive and one negative number? We’ll need to use both blue and red counters as well as some neutral pairs. If we don’t have the number of counters needed to take away, we add neutral pairs. Adding a neutral pair does not change the value. It is like changing quarters to nickels—the value is the same, but it looks different.

Let’s look at \(−5−3\) and \(5−(−3)\).

EXAMPLE \(\PageIndex{10}\)

Subtract: ⓐ \(3−1\) ⓑ \(−3−(−1)\) ⓒ \(−3−1\) ⓓ \(3−(−1)\).

EXAMPLE \(\PageIndex{11}\)

Subtract: ⓐ \(6−4\) ⓑ \(−6−(−4)\) ⓒ \(−6−4\) ⓓ \(6−(−4)\).

ⓐ \(2\) ⓑ \(−2\) ⓒ \(−10\) ⓓ \(10\)

EXAMPLE \(\PageIndex{12}\)

Subtract: ⓐ \(7−4\) ⓑ \(−7−(−4)\) ⓒ \(−7−4\) ⓓ \(7−(−4)\).

ⓐ \(3\) ⓑ \(−3\) ⓒ \(−11\) ⓓ \(11\)

Have you noticed that subtraction of signed numbers can be done by adding the opposite ? In the last example, \(−3−1\) is the same as \(−3+(−1)\) and \(3−(−1)\) is the same as \(3+1\). You will often see this idea, the Subtraction Property, written as follows:

Definition: SUBTRACTION PROPERTY

\[a−b=a+(−b)\]

Subtracting a number is the same as adding its opposite.

EXAMPLE \(\PageIndex{13}\)

Simplify: ⓐ \(13−8\) and \(13+(−8)\) ⓑ \(−17−9\) and \(−17+(−9)\) ⓒ \(9−(−15)\) and \(9+15\) ⓓ \(−7−(−4)\) and \(−7+4\).

\(\begin{array}{lccc} \text{} & −17−9 & \text{and} & −17+(−9) \\ \text{Subtract.} & −26 & \text{} & −26 \end{array}\)

\(\begin{array}{lccc} \text{} & 9−(−15) & \text{and} & 9+15 \\ \text{Subtract.} & 24 & \text{} & 24 \end{array}\)

\(\begin{array}{lccc} \text{} & −7−(−4) & \text{and} & −7+4 \\ \text{Subtract.} & −3 & \text{} & −3 \end{array}\)

EXAMPLE \(\PageIndex{14}\)

Simplify: ⓐ \(21−13\) and \(21+(−13)\) ⓑ \(−11−7\) and \(−11+(−7)\) ⓒ \(6−(−13)\) and \(6+13\) ⓓ \(−5−(−1)\) and \(−5+1\).

ⓐ \(8,8\) ⓑ \(−18,−18\)

ⓒ \(19,19\) ⓓ \(−4,−4\)

EXAMPLE \(\PageIndex{15}\)

Simplify: ⓐ \(15−7\) and \(15+(−7)\) ⓑ \(−14−8\) and \(−14+(−8)\) ⓒ \(4−(−19)\) and \(4+19\) ⓓ \(−4−(−7)\) and \(−4+7\).

ⓐ \(8,8\) ⓑ \(−22,−22\)

ⓒ \(23,23\) ⓓ \(3,3\)

What happens when there are more than three integers? We just use the order of operations as usual.

EXAMPLE \(\PageIndex{16}\)

Simplify: \(7−(−4−3)−9.\)

\(\begin{array}{lc} \text{} & 7−(−4−3)−9 \\ \text{Simplify inside the parentheses first.} & 7−(−7)−9 \\ \text{Subtract left to right.} & 14−9 \\ \text{Subtract.} & 5 \end{array}\)

Simplify: \(8−(−3−1)−9.\)

EXAMPLE \(\PageIndex{18}\)

Simplify: \(12−(−9−6)−14.\)

Multiply and Divide Integers

Since multiplication is mathematical shorthand for repeated addition, our model can easily be applied to show multiplication of integers. Let’s look at this concrete model to see what patterns we notice. We will use the same examples that we used for addition and subtraction. Here, we are using the model just to help us discover the pattern.

We remember that a⋅ba·b means add a , b times .

The next two examples are more interesting. What does it mean to multiply 5 by −3? It means subtract 5,3 times. Looking at subtraction as “taking away”, it means to take away 5, 3 times. But there is nothing to take away, so we start by adding neutral pairs on the workspace.

In summary:

\[\begin{array}{ll} 5·3=15 & −5(3)=−15 \\ 5(−3)=−15 & (−5)(−3)=15 \end{array}\]

Notice that for multiplication of two signed numbers, when the

\[ \text{signs are the } \textbf{same} \text{, the product is } \textbf{positive.} \\ \text{signs are } \textbf{different} \text{, the product is } \textbf{negative.} \]

What about division? Division is the inverse operation of multiplication. So, \(15÷3=5\) because \(15·3=15\). In words, this expression says that 15 can be divided into 3 groups of 5 each because adding five three times gives 15. If you look at some examples of multiplying integers, you might figure out the rules for dividing integers.

\[\begin{array}{lclrccl} 5·3=15 & \text{so} & 15÷3=5 & \text{ } −5(3)=−15 & \text{so} & −15÷3=−5 \\ (−5)(−3)=15 & \text{so} & 15÷(−3)=−5 & \text{ } 5(−3)=−15 & \text{so} & −15÷(−3)=5 \end{array}\]

Division follows the same rules as multiplication with regard to signs.

MULTIPLICATION AND DIVISION OF SIGNED NUMBERS

For multiplication and division of two signed numbers:

If the signs are the same, the result is positive.

If the signs are different, the result is negative.

EXAMPLE \(\PageIndex{19}\)

Multiply or divide: ⓐ \(−100÷(−4)\) ⓑ \(7⋅6\) ⓒ \(4(−8)\) ⓓ \(−27÷3.\)

\(\begin{array}{lc} \text{} & −100÷(−4) \\ \text{Divide, with signs that are} \\ \text{the same the quotient is positive.} & 25 \end{array}\)

\(\begin{array} {lc} \text{} & 7·6 \\ \text{Multiply, with same signs.} & 42 \end{array}\)

\(\begin{array} {lc} \text{} & 4(−8) \\ \text{Multiply, with different signs.} & −32 \end{array}\)

\(\begin{array}{lc} \text{} & −27÷3 \\ \text{Divide, with different signs,} \\ \text{the quotient is negative.} & −9 \end{array}\)

EXAMPLE \(\PageIndex{20}\)

Multiply or divide: ⓐ \(−115÷(−5)\) ⓑ \(5⋅12\) ⓒ \(9(−7)\) ⓓ\(−63÷7.\)

ⓐ 23 ⓑ 60 ⓒ −63 ⓓ −9

Multiply or divide: ⓐ \(−117÷(−3)\) ⓑ \(3⋅13\) ⓒ \(7(−4)\) ⓓ\(−42÷6\).

ⓐ 39 ⓑ 39 ⓒ −28 ⓓ −7

When we multiply a number by 1, the result is the same number. Each time we multiply a number by −1, we get its opposite!

MULTIPLICATION BY −1

\[−1a=−a\]

Multiplying a number by \(−1\) gives its opposite.

Simplify Expressions with Integers

What happens when there are more than two numbers in an expression? The order of operations still applies when negatives are included. Remember Please Excuse My Dear Aunt Sally?

Let’s try some examples. We’ll simplify expressions that use all four operations with integers—addition, subtraction, multiplication, and division. Remember to follow the order of operations.

EXAMPLE \(\PageIndex{22}\)

Simplify: ⓐ \((−2)^4\) ⓑ \(−2^4\).

Notice the difference in parts (a) and (b). In part (a), the exponent means to raise what is in the parentheses, the −2 to the 4 th power. In part (b), the exponent means to raise just the 2 to the 4 th power and then take the opposite.

\(\begin{array}{lc} \text{} & −2^4 \\ \text{Write in expanded form.} & −(2·2·2·2) \\ \text{We are asked to find} & \text{} \\ \text{the opposite of }24. & \text{} \\ \text{Multiply.} & −(4·2·2) \\ \text{Multiply.} & −(8·2) \\ \text{Multiply.} & −16 \end{array}\)

Simplify: ⓐ \((−3)^4\) ⓑ \(−3^4\).

ⓐ 81 ⓑ −81

EXAMPLE \(\PageIndex{24}\)

Simplify: ⓐ \((−7)^2\) ⓑ \(−7^2\).

ⓐ 49 ⓑ −49

The last example showed us the difference between \((−2)^4\) and \(−2^4\). This distinction is important to prevent future errors. The next example reminds us to multiply and divide in order left to right.

EXAMPLE \(\PageIndex{25}\)

Simplify: ⓐ \(8(−9)÷(−2)^3\) ⓑ \(−30÷2+(−3)(−7)\).

\(\begin{array}{lc} \text{} & 8(−9)÷(−2)^3 \\ \text{Exponents first.} & 8(−9)÷(−8) \\ \text{Multiply.} & −72÷(−8) \\ \text{Divide.} & 9 \end{array}\)

\(\begin{array}{lc} \text{} & −30÷2+(−3)(−7) \\ \text{Multiply and divide} \\ \text{left to right, so divide first.} & −15+(−3)(−7) \\ \text{Multiply.} & −15+21 \\ \text{Add.} & 6 \end{array}\)

Simplify: ⓐ \(12(−9)÷(−3)^3\) ⓑ \(−27÷3+(−5)(−6).\)

ⓐ 4 ⓑ 21

EXAMPLE \(\PageIndex{27}\)

Simplify: ⓐ \(18(−4)÷(−2)^3\) ⓑ \(−32÷4+(−2)(−7).\)

ⓐ 9 ⓑ 6

Evaluate Variable Expressions with Integers

Remember that to evaluate an expression means to substitute a number for the variable in the expression. Now we can use negative numbers as well as positive numbers.

EXAMPLE \(\PageIndex{28}\)

Evaluate \(4x^2−2xy+3y^2\) when \(x=2,y=−1\).

EXAMPLE \(\PageIndex{29}\)

Evaluate: \(3x^2−2xy+6y^2\) when \(x=1,y=−2\).

EXAMPLE \(\PageIndex{30}\)

Evaluate: \(4x^2−xy+5y^2\) when \(x=−2,y=3\).

Translate Phrases to Expressions with Integers

Our earlier work translating English to algebra also applies to phrases that include both positive and negative numbers.

EXAMPLE \(\PageIndex{31}\)

Translate and simplify: the sum of 8 and −12, increased by 3.

\(\begin{array}{lc} \text{} & \text{the } \textbf{sum } \underline{\text{of}} \; –8 \; \underline{\text{and}} −12 \text{ increased by } 3 \\ \text{Translate.} & [8+(−12)]+3 \\ \text{Simplify. Be careful not to confuse the} \; \; \; \; \; \; \; \; \; \; & (−4)+3 \\ \text{brackets with an absolute value sign.} \\ \text{Add.} & −1 \end{array}\)

EXAMPLE \(\PageIndex{32}\)

Translate and simplify the sum of 9 and −16, increased by 4.

\((9+(−16))+4;−3\)

EXAMPLE \(\PageIndex{33}\)

Translate and simplify the sum of −8 and −12, increased by 7.

\((−8+(−12))+7;−13\)

Use Integers in Applications

We’ll outline a plan to solve applications. It’s hard to find something if we don’t know what we’re looking for or what to call it! So when we solve an application, we first need to determine what the problem is asking us to find. Then we’ll write a phrase that gives the information to find it. We’ll translate the phrase into an expression and then simplify the expression to get the answer. Finally, we summarize the answer in a sentence to make sure it makes sense.

EXAMPLE \(\PageIndex{34}\): How to Solve Application Problems Using Integers

The temperature in Kendallville, Indiana one morning was 11 degrees. By mid-afternoon, the temperature had dropped to −9−9degrees. What was the difference in the morning and afternoon temperatures?

EXAMPLE \(\PageIndex{35}\)

The temperature in Anchorage, Alaska one morning was 15 degrees. By mid-afternoon the temperature had dropped to 30 degrees below zero. What was the difference in the morning and afternoon temperatures?

The difference in temperatures was 45 degrees Fahrenheit.

EXAMPLE \(\PageIndex{36}\)

The temperature in Denver was −6 degrees at lunchtime. By sunset the temperature had dropped to −15 degrees. What was the difference in the lunchtime and sunset temperatures?

The difference in temperatures was 9 degrees.

USE INTEGERS IN APPLICATIONS.

• Read the problem. Make sure all the words and ideas are understood.
• Identify what we are asked to find.
• Write a phrase that gives the information to find it.
• Translate the phrase to an expression.
• Simplify the expression.
• Answer the question with a complete sentence.

Access this online resource for additional instruction and practice with integers.

• Subtracting Integers with Counters

Key Concepts

• \[\begin{align} & −a \text{ means the opposite of the number }a \\ & \text{The notation} −a \text{ is read as “the opposite of }a \text{.”} \end{align} \]

The absolute value of a number n is written as \(|n|\) and \(|n|≥0\) for all numbers.

• Subtraction Property \(a−b=a+(−b)\) Subtracting a number is the same as adding its opposite.

\(−1a=−a\)

• Read the problem. Make sure all the words and ideas are understood

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• Jovan Fonte
• Matin Naseri
• Sharky Kesa
• Anuj Shikarkhane

An integer is a number that does not have a fractional part. The set of integers is

\[\mathbb{Z}=\{\cdots -4, -3, -2, -1, 0, 1, 2, 3, 4 \dots\}. \]

The notation \(\mathbb{Z}\) for the set of integers comes from the German word Zahlen , which means "numbers". Integers strictly larger than zero are positive integers and integers strictly less than zero are negative integers .

For example, \(2\), \(67\), \(0\), and \(-13\) are all integers (2 and 67 are positive integers and -13 is a negative integer). The values \(\frac{4}{7}\), \(10.7\), \(\frac{34}{7}\), \(\sqrt{2}\), and \(\pi\) are not integers.

Properties of Integers

Integers - problem solving.

The following are the properties of integers:

• The set of integers is closed under the operation of addition: if \(a, b \in \mathbb{Z}\), then \(a+b \in \mathbb{Z}\).
• The set of integers is closed under the operation of multiplication: if \(a, b \in \mathbb{Z}\), then \(ab\in \mathbb{Z}\).
• For any integer \(a\), the additive inverse \(-a\) is an integer.
• If \(a\) and \(b\) are integers such that \(a \cdot b = 0\), then \(a=0\) or \(b=0\).
• The set of integers is infinite and has no smallest element and no largest element.

\((\in\) means "belongs to", as \(a \in Z\) means \(a\) is an element of the set \(Z\) or \(a\) belongs to the set \(Z.)\)

Note that the set of integers is not closed under the operation of division. As an example, \(a=3\) and \(b=4\) are integers, but \(\frac{a}{b} = \frac{3}{4} \) is not an integer.

Which of the following are integers? \[\begin{array} &\frac{4}{2}, &-8, &0. 2, &12-4, &\frac{10}{4} \end{array}\] Since \(\frac{4}{2} = 2\), \(12-4 = 8,\) and \(2 < \frac{10}{4} < 3\), the integers in the list are \( \frac{4}{2}, -8 \), and \( 12-4.\) \(_\square\)

What is the smallest integer that is larger than \(\frac{10}{3}?\) Since \( 3< \frac{10}{3} < 4,\) the smallest integer that is larger than \(\frac{10}{3}\) is \(4\). \(_\square\)

What is the largest integer that is smaller than \(\frac{10}{3}?\) Since \( 3< \frac{10}{3} < 4,\) the largest integer that is smaller than \(\frac{10}{3}\) is \(3\). \(_\square\)

Using the properties of integers above, show that set of integers is closed under the operation of subtraction. Consider any two integers \(a\) and \(b\). We would like to show \(a-b\) is also an integer. By property \(3,\) the additive inverse of \(b\) is \(-b\), which is an integer. Then \[ a-b = a + (-b)\] is an integer by Property \(1.\) Therefore, the set of integers is closed under the operation of subtraction. \(_\square\)

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Course: Class 6   >   Unit 5

• Subtracting a negative = adding a positive
• Subtracting integers: find the missing value
• Understand subtraction as adding the opposite
• Subtracting negative numbers

Addition and subtraction of integers

• Your answer should be
• an integer, like 6 ‍  
• a simplified proper fraction, like 3 / 5 ‍  
• a simplified improper fraction, like 7 / 4 ‍  
• a mixed number, like 1   3 / 4 ‍  
• an exact decimal, like 0.75 ‍  
• a multiple of pi, like 12   pi ‍   or 2 / 3   pi ‍