Subtract Integers

Model Subtraction of IntegersRemember the story in the last section about the toddler and the cookies? Children learn how to subtract numbers through their everyday experiences. Real-life experiences serve as models for subtracting positive numbers, and in some cases, such as temperature, for adding negative as well as positive numbers. But it is difficult to relate subtracting negative numbers to common life experiences. Most people do not have an intuitive understanding of subtraction when negative numbers are involved. Math teachers use several different models to explain subtracting negative numbers. We will continue to use counters to model subtraction. Remember, the blue counters represent positive numbers and the red counters represent negative numbers. Perhaps when you were younger, you read 5−3 as five take away three. When we use counters, we can think of subtraction the same way. Doing the Manipulative Mathematics activity "Subtraction of Signed Numbers" will help you develop a better understanding of subtracting integers.We will model four subtraction facts using the numbers 5 and 3. 5−3−5−(−3)−5−35−(−3) Model: 5−3. Solution Interpret the expression. 5−3 means 5 take away 3. Model the first number. Start with 5 positives.

Take away the second number. So take away 3 positives.

Find the counters that are left.

5−3=2.The difference between 5 and 3 is 2.

Model the expression:6−4

2 Model the expression:7−4

3 Model: −5−(−3). Solution Interpret the expression. −5−(−3) means −5 take away −3. Model the first number. Start with 5 negatives.

Take away the second number. So take away 3 negatives.

Find the number of counters that are left.

−5−(−3)=−2.The difference between −5 and −3 is −2.

Model the expression:−6−(−4)

−2 Model the expression:−7−(−4)

−3 Notice that and are very much alike. First, we subtracted 3 positives from 5 positives to get 2 positives. Then we subtracted 3 negatives from 5 negatives to get 2 negatives. Each example used counters of only one color, and the “take away” model of subtraction was easy to apply.

Now let’s see what happens when we subtract one positive and one negative number. We will need to use both positive and negative counters and sometimes some neutral pairs, too. Adding a neutral pair does not change the value. Model: −5−3. Solution Interpret the expression. −5−3 means −5 take away 3. Model the first number. Start with 5 negatives.

Take away the second number.So we need to take away 3 positives. But there are no positives to take away.Add neutral pairs until you have 3 positives.

Now take away 3 positives.

Count the number of counters that are left.

−5−3=−8.The difference of −5 and 3 is −8.

Model the expression:−6−4

−10 Model the expression:−7−4

−11 Model: 5−(−3). Solution Interpret the expression. 5−(−3) means 5 take away −3. Model the first number. Start with 5 positives.

Take away the second number, so take away 3 negatives. But there are no negatives to take away.Add neutral pairs until you have 3 negatives.

Then take away 3 negatives.

Count the number of counters that are left.

The difference of 5 and −3 is 8.5−(−3)=8

Model the expression:6−(−4)

10 Model the expression:7−(−4)

11 Model each subtraction. ⓐ 8 − 2 ⓑ −5 − 4 ⓒ 6 − (−6) ⓓ −8 − (−3) ⓐ 8−2This means 8 take away 2. Start with 8 positives.

Take away 2 positives.

How many are left? 6 8−2=6

ⓑ −5−4This means −5 take away 4. Start with 5 negatives.

You need to take away 4 positives.Add 4 neutral pairs to get 4 positives.

Take away 4 positives.

How many are left?

−9 −5−4=−9

Add 6 neutrals to get 6 negatives to take away.

Remove 6 negatives.

How many are left?

12 6−(−6)=12

ⓓ −8−(−3)This means −8 take away −3. Start with 8 negatives.

Take away 3 negatives.

How many are left?

−5 −8−(−3)=−5

Model each subtraction. ⓐ 7 - (-8) ⓑ -7 - (-2) ⓒ 4 - 1 ⓓ -6 - 8 ⓐ

ⓑ

ⓒ

ⓓ

Model each subtraction. ⓐ 4 - (-6) ⓑ -8 - (-1) ⓒ 7 - 3 ⓓ -4 - 2 ⓐ

ⓑ

ⓒ

ⓓ

Model each subtraction expression: ⓐ 2−8 ⓑ −3−(−8) Solution ⓐ We start with 2 positives.

We need to take away 8 positives, but we have only 2. Add neutral pairs until there are 8 positives to take away.

Then take away eight positives.

Find the number of counters that are left.There are 6 negatives.

2−8=−6

ⓑ We start with 3 negatives.

We need to take away 8 negatives, but we have only 3. Add neutral pairs until there are 8 negatives to take away.

Then take away the 8 negatives.

Find the number of counters that are left.There are 5 positives.

−3−(−8)=5

Model each subtraction expression. ⓐ 7−9 ⓑ −5−(−9) ⓐ

−2 ⓑ

4 Model each subtraction expression. ⓐ 4−7 ⓑ −7−(−10) ⓐ

−3 ⓑ

Simplify Expressions with Integers Do you see a pattern? Are you ready to subtract integers without counters? Let’s do two more subtractions. We’ll think about how we would model these with counters, but we won’t actually use the counters. Subtract −23−7. Think: We start with 23 negative counters. We have to subtract 7 positives, but there are no positives to take away. So we add 7 neutral pairs to get the 7 positives. Now we take away the 7 positives. So what’s left? We have the original 23 negatives plus 7 more negatives from the neutral pair. The result is 30 negatives. −23−7=−30 Notice, that to subtract 7, we added 7 negatives. Subtract 30−(−12). Think: We start with 30 positives. We have to subtract 12 negatives, but there are no negatives to take away. So we add 12 neutral pairs to the 30 positives. Now we take away the 12 negatives. What’s left? We have the original 30 positives plus 12 more positives from the neutral pairs. The result is 42 positives. 30−(−12)=42 Notice that to subtract −12, we added 12. While we may not always use the counters, especially when we work with large numbers, practicing with them first gave us a concrete way to apply the concept, so that we can visualize and remember how to do the subtraction without the counters. Have you noticed that subtraction of signed numbers can be done by adding the opposite? You will often see the idea, the Subtraction Property, written as follows: Subtraction Propertya−b=a+(−b)Look at these two examples.

We see that 6−4 gives the same answer as 6+(−4). Of course, when we have a subtraction problem that has only positive numbers, like the first example, we just do the subtraction. We already knew how to subtract 6−4 long ago. But knowing that 6−4 gives the same answer as 6+(−4) helps when we are subtracting negative numbers. Simplify: ⓐ 13−8and13+(−8) ⓑ −17−9and−17+(−9) Solution ⓐ 13−8 and 13+(−8) Subtract to simplify. 13−8=5 Add to simplify. 13+(−8)=5 Subtracting 8 from 13 is the same as adding −8 to 13.

ⓑ −17−9 and −17+(−9) Subtract to simplify. −17−9=−26 Add to simplify. −17+(−9)=−26 Subtracting 9 from −17 is the same as adding −9 to −17.

Simplify each expression: ⓐ 21−13and21+(−13) ⓑ −11−7and−11+(−7) ⓐ 8, 8 ⓑ −18, −18 Simplify each expression: ⓐ 15−7and15+(−7) ⓑ −14−8and−14+(−8) ⓐ 8, 8 ⓑ −22, −22 Now look what happens when we subtract a negative.

We see that 8−(−5) gives the same result as 8+5. Subtracting a negative number is like adding a positive. Simplify: ⓐ 9−(−15)and9+15 ⓑ −7−(−4)and−7+4 Solution ⓐ 9−(−15) and 9+15 Subtract to simplify. 9−(−15)=24 Add to simplify. 9+15=24 Subtracting −15 from 9 is the same as adding 15 to 9.

ⓑ −7−(−4) and −7+4 Subtract to simplify. −7−(−4)=−3 Add to simplify. −7+4=−3 Subtracting −4 from −7 is the same as adding 4 to −7

Simplify each expression: ⓐ 6−(−13)and6+13 ⓑ −5−(−1)and−5+1 ⓐ 19, 19 ⓑ −4, −4 Simplify each expression: ⓐ 4−(−19)and4+19 ⓐ −4−(−7)and−4+7 ⓐ 23, 23 ⓑ 3, 3 Look again at the results of - .Subtraction of Integers 5–3 –5–(–3) 2 –2 2 positives 2 negatives When there would be enough counters of the color to take away, subtract. –5–3 5–(–3) –8 8 5 negatives, want to subtract 3 positives 5 positives, want to subtract 3 negatives need neutral pairs need neutral pairs When there would not be enough of the counters to take away, add neutral pairs.

Simplify: −74−(−58). Solution We are taking 58 negatives away from 74 negatives. −74−(−58) Subtract. −16

Simplify the expression: −67−(−38) −29 Simplify the expression: −83−(−57) −26 Simplify: 7−(−4−3)−9. Solution We use the order of operations to simplify this expression, performing operations inside the parentheses first. Then we subtract from left to right. Simplify inside the parentheses first.

Subtract from left to right.

Subtract.

Simplify the expression: 8−(−3−1)−9 3 Simplify the expression: 12−(−9−6)−14 13 Simplify: 3·7−4·7−5·8. Solution We use the order of operations to simplify this expression. First we multiply, and then subtract from left to right. Multiply first.

Subtract from left to right.

Subtract.

Simplify the expression: 6·2−9·1−8·9. −69 Simplify the expression: 2·5−3·7−4·9 −47

Subtract Integers in ApplicationsIt’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 problem, we first need to determine what we are asked to find. Then we can 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. Solve Application Problems.Identify what you 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. In the morning, the temperature in Urbana, Illinois was 11 degrees Fahrenheit. By mid-afternoon, the temperature had dropped to −9 degrees Fahrenheit. What was the difference between the morning and afternoon temperatures? Solution Step 1. Identify what we are asked to find. the difference between the morning and afternoon temperatures Step 2. Write a phrase that gives the information to find it. the difference of 11 and −9 Step 3. Translate the phrase to an expression.The word difference indicates subtraction. 11−(−9) Step 4. Simplify the expression. 20 Step 5. Write a complete sentence that answers the question. The difference in temperature was 20 degrees Fahrenheit.

In the morning, the temperature in Anchorage, Alaska was 15 degrees Fahrenheit. By mid-afternoon the temperature had dropped to 30 degrees below zero. What was the difference between the morning and afternoon temperatures? 45 degrees Fahrenheit The temperature in Denver was −6 degrees Fahrenheit at lunchtime. By sunset the temperature had dropped to −15 degree Fahrenheit. What was the difference between the lunchtime and sunset temperatures? 9 degrees Fahrenheit Geography provides another application of negative numbers with the elevations of places below sea level. Dinesh hiked from Mt. Whitney, the highest point in California, to Death Valley, the lowest point. The elevation of Mt. Whitney is 14,497 feet above sea level and the elevation of Death Valley is 282 feet below sea level. What is the difference in elevation between Mt. Whitney and Death Valley? Solution Step 1. What are we asked to find? The difference in elevation between Mt. Whitney and Death Valley Step 2. Write a phrase. elevation of Mt. Whitney−elevation of Death Valley Step 3. Translate. 14,497−(−282) Step 4. Simplify. 14,779 Step 5. Write a complete sentence that answers the question. The difference in elevation is 14,779 feet.

One day, John hiked to the 10,023 foot summit of Haleakala volcano in Hawaii. The next day, while scuba diving, he dove to a cave 80 feet below sea level. What is the difference between the elevation of the summit of Haleakala and the depth of the cave? 10,103 feet The submarine Nautilus is at 340 feet below the surface of the water and the submarine Explorer is 573 feet below the surface of the water. What is the difference in the position of the Nautilus and the Explorer? 233 feet Managing your money can involve both positive and negative numbers. You might have overdraft protection on your checking account. This means the bank lets you write checks for more money than you have in your account (as long as they know they can get it back from you!) Leslie has $25 in her checking account and she writes a check for $8. ⓐ What is the balance after she writes the check? ⓑ She writes a second check for $20. What is the new balance after this check? ⓒ Leslie’s friend told her that she had lost a check for $10 that Leslie had given her with her birthday card. What is the balance in Leslie’s checking account now? Solution ⓐ What are we asked to find? The balance of the account Write a phrase. $25 minus $8 Translate

Simplify.

Write a sentence answer. The balance is $17.

ⓑ What are we asked to find? The new balance Write a phrase. $17 minus $20 Translate

Simplify.

Write a sentence answer. She is overdrawn by $3.

Simplify.

Write a sentence answer. The balance is now $7.

Araceli has $75 in her checking account and writes a check for $27. ⓐ What is the balance after she writes the check? ⓑ She writes a second check for $50. What is the new balance? ⓒ The check for $20 that she sent a charity was never cashed. What is the balance in Araceli’s checking account now? ⓐ $48 ⓑ −$2 ⓒ $18 Genevieve’s bank account was overdrawn and the balance is −$78. ⓐ She deposits a check for $24 that she earned babysitting. What is the new balance? ⓑ She deposits another check for $49. Is she out of debt yet? What is her new balance? ⓐ −$54 ⓑ No, −$5 The Links to Literacy activity "Elevator Magic" will provide you with another view of the topics covered in this section. ACCESS ADDITIONAL ONLINE RESOURCES Adding and Subtracting Integers Subtracting Integers with Color Counters Subtracting Integers Basics Subtracting Integers Integer Application

Practice Makes PerfectModel Subtraction of Integers In the following exercises, model each expression and simplify. 8−2

6 9−3 −5−(−1)

−4 −6−(−4) −5−4

−9 −7−2 8−(−4)

12 7−(−3) Simplify Expressions with Integers In the following exercises, simplify each expression. ⓐ 15−6 ⓑ 15+(−6) ⓐ 9 ⓑ 9 ⓐ 12−9 ⓑ 12+(−9) ⓐ 44−28 ⓑ 44+(−28) ⓐ 16 ⓑ 16 ⓐ 35−16 ⓑ 35+(−16) ⓐ 8−(−9) ⓑ 8+9 ⓐ 17 ⓑ 17 ⓐ 4−(−4) ⓑ 4+4 ⓐ 27−(−18) ⓑ 27+18 ⓐ 45 ⓑ 45 ⓐ 46−(−37) ⓑ 46+37 In the following exercises, simplify each expression. 15−(−12) 27 14−(−11) 10−(−19) 29 11−(−18) 48−87 −39 45−69 31−79 −48 39−81 −31−11 −42 −32−18 −17−42 −59 −19−46 −103−(−52) −51 −105−(−68) −45−(−54) 9 −58−(−67) 8−3−7 −2 9−6−5 −5−4+7 −2 −3−8+4 −14−(−27)+9 22 −15−(−28)+5 71+(−10)−8 53 64+(−17)−9 −16−(−4+1)−7 −20 −15−(−6+4)−3 (2−7)−(3−8) 0 (1−8)−(2−9) −(6−8)−(2−4) 4 −(4−5)−(7−8) 25−[10−(3−12)] 6 32−[5−(15−20)] 6⋅3−4⋅3−7⋅2 –8 5⋅7−8⋅2−4⋅9 52−62 −11 62−72 Evaluate Variable Expressions with Integers In the following exercises, evaluate each expression for the given values. x−6when ⓐ x=3 ⓑ x=−3 ⓐ −3 ⓑ −9 x−4when ⓐ x=5 ⓑ x=−5 5−ywhen ⓐ y=2 ⓑ y=−2 ⓐ 3 ⓑ 7 8−ywhen ⓐ y=3 ⓑ y=−3 4x2−15x+1whenx=3 −8 5x2−14x+7whenx=2 −12−5x2whenx=6 −192 −19−4x2whenx=5 Translate Word Phrases to Algebraic Expressions In the following exercises, translate each phrase into an algebraic expression and then simplify. ⓐ The difference of 3 and −10 ⓑ Subtract −20 from 45 ⓐ 3 − (−10) = 13 ⓑ 45 − (−20) = 65 ⓐ The difference of 8 and −12 ⓑ Subtract −13 from 50 ⓐ The difference of −6 and 9 ⓑ Subtract −12 from −16 ⓐ −6 − 9 = −15 ⓑ −16 − (−12) = −4 ⓐ The difference of −8 and 9 ⓑ Subtract −15 from −19 ⓐ 8 less than −17 ⓑ −24 minus 37 ⓐ −17 − 8 = −25 ⓑ −24 − 37 = −61 ⓐ 5 less than −14 ⓑ −13 minus 42 ⓐ 21 less than6 ⓑ 31 subtracted from −19 ⓐ 6 − 21 = −15 ⓑ −19 − 31 = −50 ⓐ 34 less than7 ⓑ 29 subtracted from −50 Subtract Integers in Applications In the following exercises, solve the following applications. Temperature One morning, the temperature in Urbana, Illinois, was 28° Fahrenheit. By evening, the temperature had dropped 38° Fahrenheit. What was the temperature that evening? −10° Temperature On Thursday, the temperature in Spincich Lake, Michigan, was 22° Fahrenheit. By Friday, the temperature had dropped 35° Fahrenheit. What was the temperature on Friday? Temperature On January 15, the high temperature in Anaheim, California, was 84° Fahrenheit. That same day, the high temperature in Embarrass, Minnesota was −12° Fahrenheit. What was the difference between the temperature in Anaheim and the temperature in Embarrass? 96° Temperature On January 21, the high temperature in Palm Springs, California, was 89°, and the high temperature in Whitefield, New Hampshire was −31°. What was the difference between the temperature in Palm Springs and the temperature in Whitefield? Football At the first down, the Warriors football team had the ball on their 30-yard line. On the next three downs, they gained 2 yards, lost 7 yards, and lost 4 yards. What was the yard line at the end of the third down? 21-yard line Football At the first down, the Barons football team had the ball on their 20-yard line. On the next three downs, they lost 8 yards, gained 5 yards, and lost 6 yards. What was the yard line at the end of the third down? Checking Account John has $148 in his checking account. He writes a check for $83. What is the new balance in his checking account? $65 Checking Account Ellie has $426 in her checking account. She writes a check for $152. What is the new balance in her checking account? Checking Account Gina has $210 in her checking account. She writes a check for $250. What is the new balance in her checking account? −$40 Checking Account Frank has $94 in his checking account. He writes a check for $110. What is the new balance in his checking account? Checking Account Bill has a balance of −$14 in his checking account. He deposits $40 to the account. What is the new balance? $26 Checking Account Patty has a balance of −$23 in her checking account. She deposits $80 to the account. What is the new balance?

Everyday Math Camping Rene is on an Alpine hike. The temperature is−7°. Rene’s sleeping bag is rated “comfortable to −20°”. How much can the temperature change before it is too cold for Rene’s sleeping bag? 13° Scuba Diving Shelly’s scuba watch is guaranteed to be watertight to −100feet. She is diving at −45feet on the face of an underwater canyon. By how many feet can she change her depth before her watch is no longer guaranteed?

Writing Exercises Explain why the difference of 9 and −6 is 15. Sample answer: On a number line, 9 is 15 units away from −6. Why is the result of subtracting 3−(−4) the same as the result of adding 3+4?

Self Checkⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?