Fundamentals Of Mathematics I

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Fundamentals of Mathematics IKent State Department of Mathematical SciencesFall 2008Available ugust 4, 2008

Contents1 Arithmetic1.1 Real Numbers . . . . . . . . . . . .1.1.1 Exercises 1.1 . . . . . . . .1.2 Addition . . . . . . . . . . . . . .1.2.1 Exercises 1.2 . . . . . . . .1.3 Subtraction . . . . . . . . . . . . .1.3.1 Exercises 1.3 . . . . . . . .1.4 Multiplication . . . . . . . . . . .1.4.1 Exercises 1.4 . . . . . . . .1.5 Division . . . . . . . . . . . . . . .1.5.1 Exercise 1.5 . . . . . . . . .1.6 Exponents . . . . . . . . . . . . .1.6.1 Exercises 1.6 . . . . . . . .1.7 Order of Operations . . . . . . . .1.7.1 Exercises 1.7 . . . . . . . .1.8 Primes, Divisibility, Least Common1.8.1 Exercises 1.8 . . . . . . . .1.9 Fractions and Percents . . . . . . .1.9.1 Exercises 1.9 . . . . . . . .1.10 Introduction to Radicals . . . . . .1.10.1 Exercises 1.10 . . . . . . . .1.11 Properties of Real Numbers . . . .1.11.1 Exercises 1.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Denominator, Greatest Common. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Factor. . . . . . . . . . . . . . . . . . . . . .2277121219192323282831313334404050515353572 Basic Algebra2.1 Combining Like Terms . . . . . . .2.1.1 Exercises 2.1 . . . . . . . .2.2 Introduction to Solving Equations2.2.1 Exercises 2.2 . . . . . . . .2.3 Introduction to Problem Solving .2.3.1 Exercises 2.3 . . . . . . . .2.4 Computation with Formulas . . . .2.4.1 Exercises 2.4 . . . . . . . .585860606566717176.3 Solutions to Exercises.771

Chapter 1Arithmetic1.1Real NumbersAs in all subjects, it is important in mathematics that when a word is used, an exact meaning needs to be properlyunderstood. This is where we will begin.When you were young an important skill was to be able to count your candy to make sure your sibling did not cheat youout of your share. These numbers can be listed: {1, 2, 3, 4, .}. They are called counting numbers or positive integers.When you ran out of candy you needed another number 0. This set of numbers can be listed {0, 1, 2, 3, .}. They arecalled whole numbers or non-negative integers. Note that we have used set notation for our list. A set is just acollection of things. Each thing in the collection is called an element or member the set. When we describea set by listing its elements, we enclose the list in curly braces, ‘{}’. In notation {1, 2, 3, .}, the ellipsis, ‘.’,means that the list goes on forever in the same pattern. So for example, we say that the number 23 is anelement of the set of positive integers because it will occur on the list eventually. Using the language of sets,we say that 0 is an element of the non-negative integers but 0 is not an element of the positive integers. Wealso say that the set of non-negative integers contains the set of positive integers.As you grew older, you learned the importance of numbers in measurements. Most people check the temperature beforethey leave their home for the day. In the summer we often estimate to the nearest positive integer (choose the closestcounting number). But in the winter we need numbers that represent when the temperature goes below zero. We canestimate the temperature to numbers in the set {., 3, 2, 1, 0, 1, 2, 3, .}. These numbers are called integers.The real numbers are all of the numbers that can be represented on a number line. This includes the integers labeledon the number line below. (Note that the number line does not stop at -7 and 7 but continues on in both directions asrepresented by arrows on the ends.)To plot a number on the number line place a solid circle or dot on the number line in the appropriate place.Examples: Sets of Numbers & Number LineExample 1Solution:Plot on the number line the integer -3.Practice 2Plot on the number line the integer -5.Solution: Click here to check your answer.2

Example 3Of which set(s) is 0 an element: integers, non-negative integers or positive integers?Solution: Since 0 is in the listings {0, 1, 2, 3, .} and {., 2, 1, 0, 1, 2, .} but not in {1, 2, 3, .}, it is an element of theintegers and the non-negative integers.Practice 4Of which set(s) is 5 an element: integers, non-negative integers or positive integers?Solution: Click here to check your answer.When it comes to sharing a pie or a candy bar we need numbers which represent a half, a third, or any partial amountthat we need. A fraction is an integer divided by a nonzero integer. Any number that can be written as a fraction is calleda rational number. For example, 3 is a rational number since 3 3 1 31 . All integers are rational numbers. Noticethat a fraction is nothing more than a representation of a division problem. We will explore how to convert a decimal to afraction and vice versa in section 1.9.Consider the fraction 21 . One-half of the burgandy rectangle below is the gray portion in the next picture. It representshalf of the burgandy rectangle. That is, 1 out of 2 pieces. Notice that the portions must be of equal size.Rational numbers are real numbers which can be written as a fraction and therefore can be plotted on a number line. Butthere are other real numbers which cannot be rewritten as a fraction. In order to consider this, we will discuss decimals. Ournumber system is based on 10. You can understand this when you are dealing with the counting numbers. For example, 10ones equals 1 ten, 10 tens equals 1 one-hundred and so on. When we consider a decimal, it is also based on 10. Consider thenumber line below where the red lines are the tenths, that is, the number line split up into ten equal size pieces between 0and 1. The purple lines represent the hundredths; the segment from 0 to 1 on the number line is split up into one-hundredequal size pieces between 0 and 1.As in natural numbers these decimal places have place values. The first place to the right of the decimal is the tenthsthen the hundredths. Below are the place values to the millionths.tens: ones: . : tenths: hundredths: thousandths: ten-thousandths: hundred-thousandths: millionthsThe number 13.453 can be read “thirteen and four hundred fifty-three thousandths”. Notice that after the decimalyou read the number normally adding the ending place value after you state the number. (This can be read informally as“thirteen point four five three.) Also, the decimal is indicated with the word “and”. The decimal 1.0034 would be “one andthirty-four ten-thousandths”.Real numbers that are not rational numbers are called irrational numbers. Decimals that do not terminate (end) orrepeat represent irrational numbers. The set of all rational numbers together with the set of irrational numbers is calledthe set of real numbers. The diagrambelowshows the relationship between the sets of numbers discussed so far. Some examples of irrational numbers are 2, π, 6 (radicals will be discussed further in Section 1.10). There are infinitely manyirrational numbers. The diagram below shows the terminology of the real numbers and their relationship to each other.All the sets in the diagram are real numbers. The colors indicate the separation between rational (shades of green) andirrational numbers (blue). All sets that are integers are in inside the oval labeled integers, while the whole numbers containthe counting numbers.3

Examples: Decimals on the Number LineExample 5a) Plot 0.2 on the number line with a black dot.b) Plot 0.43 with a green dot.Solution: For 0.2 we split the segment from 0 to 1 on the number line into ten equal pieces between 0 and 1 and then countover 2 since the digit 2 is located in the tenths place. For 0.43 we split the number line into one-hundred equal pieces between0 and 1 and then count over 43 places since the digit 43 is located in the hundredths place. Alternatively, we can split upthe number line into ten equal pieces between 0 and 1 then count over the four tenths. After this split the number line upinto ten equal pieces between 0.4 and 0.5 and count over 3 places for the 3 hundredths.Practice 6a) Plot 0.27 on the number line with a black dot.b) Plot 0.8 with a green dot.Solution: Click here to check your answer.Example 7a) Plot 3.16 on the number line with a black dot.b) Plot 1.62 with a green dot.Solution: a) Using the first method described for 3.16, we split the number line between the integers 3 and 4 into one hundredequal pieces and then count over 16 since the digit 16 is located in the hundredths place.4

b) Using the second method described for 1.62, we split the number line into ten equal pieces between 1 and 2 andthen count over 6 places since the digit 6 is located in the tenths place. Then split the number line up into ten equal piecesbetween 0.6 and 0.7 and count over 2 places for the 2 hundredths.Practice 8a) Plot 4.55 on the number line with a black dot.b) Plot 7.18 with a green dot.Solution: Click here to check your answer.Example 9a) Plot -3.4 on the number line with a black dot.b) Plot -3.93 with a green dot.Solution: a) For -3.4, we split the number line between the integers -4 and -3 into one ten equal pieces and then count to theleft (for negatives) 4 units since the digit 4 is located in the tenths place.b) Using the second method, we place -3.93 between -3.9 and -4 approximating the location.Practice 10a) Plot -5.9 on the number line with a black dot.b) Plot -5.72 with a green dot.Solution: Click here to check your answer.Often in real life we desire to know which is a larger amount. If there are 2 piles of cash on a table most people wouldcompare and take the pile which has the greater value. Mathematically, we need some notation to represent that 20 isgreater than 15. The sign we use is (greater than). We write, 20 15. It is worth keeping in mind a little memorytrick with these inequality signs. The thought being that the mouth always eats the larger number.This rule holds even when the smaller number comes first. We know that 2 is less than 5 and we write 2 5 where indicates“less than”. In comparison we also have the possibility of equality which is denoted by . There are two combinations thatcan also be used less than or equal to and greater than or equal to. This is applicable to our daily lives when we considerwanting “at least” what the neighbors have which would be the concept of . Applications like this will be discussed later.When some of the numbers that we are comparing might be negative, a question arises. For example, is 4 or 3greater? If you owe 4 and your friend owes 3, you have the larger debt which means you have “less” money. So, 4 3.When comparing two real numbers the one that lies further to the left on the number line is always the lesser of the two.Consider comparing the two numbers in Example 9, 3.4 and 3.93.Since 3.93 is further left than 3.4, we have that 3.4 3.93 or 3.4 3.93 are true. Similarly, if we reverse the orderthe following inequalities are true 3.93 3.4 or 3.93 3.4.Examples: Inequalities5

Example 11State whether the following are true:a) 5 4b) 4.23 4.2Solution:a) True, because 5 is further left on the number line than 4.b) False, because 4.23 is 0.03 units to the right of 4.2 making 4.2 the smaller number.Practice 12State whether the following are true:a) 10 11b) 7.01 7.1Solution: Click here to check your answer.Solutions to Practice Problems:Practice 2Back to TextPractice 4Since 5 is in the listings {0, 1, 2, 3, .}, {., 2, 1, 0, 1, 2, .} and {1, 2, 3, .}, it is an element of the non-negative integers(whole numbers), the integers and the positive integers (or counting numbers). Back to TextPractice 6Back to TextPractice 8Back to TextPractice 10Back to TextPractice 12Solution:a) 10 11 is true since 11 is further left on the number line making it the smaller number.b) 7.01 7.1 is true since 7.01 is further left on the number line making it the smaller number.Back to Text6

1.1.1Exercises 1.1Determine to which set or sets of numbers the following elements belong: irrational, rational, integers, whole numbers,positive integers. Click here to see examples.1. 132. 50 4. 3.55. 15Plot the following numbers on the number line. Click here to see examples.3. 126. 5.3337. 98. 910. 3.4711. 1.23State whether the following are true: Click here to see examples.9. 012. 5.1113. 4 416. 30.5 30.0514. 5 217. 4 415. 20 1218. 71.24 71.2Click here to see the solutions.1.2AdditionThe concept of distance from a starting point regardless of direction is important. We often go to the closest gas stationwhen we are low on gas. The absolute value of a number is the distance on the number line from zero to the numberregardless of the sign of the number. The absolute value is denoted using vertical lines # . For example, 4 4 since it isa distance of 4 on the number line from the starting point, 0. Similarly, 4 4 since it is a distance of 4 from 0. Sinceabsolute value can be thought of as the distance from 0 the resulting answer is a nonnegative number.Examples: Absolute ValueExample 1Calculate 6 Solution: 6 6 since 6 is six units from zero. This can be seen below by counting the units in red on the number line.Practice 2Calculate 11 Solution: Click here to check your answer.Notice that the absolute value only acts on a single number. You must do any arithmetic inside first.We will build on this basic understanding of absolute value throughout this course.When adding non-negative integers there are many ways to consider the meaning behind adding. We will take a lookat two models which will help us understand the meaning of addition for integers.The first model is a simple counting example. If we are trying to calculate 13 14, we can gather two sets of objects,one with 13 and one containing 14. Then count all the objects for the answer. (See picture below.)7

If there are thirteen blue boxes in one corner and fourteen blue boxes in another corner altogether there are 27 blue boxes.The mathematical sentence which represents this problem is 13 14 27.Another way of considering addition of positive integers is by climbing steps. Consider taking one step and then twomore steps, altogether you would take 3 steps. The mathematical sentence which represents this problem is 1 2 3.Even though the understanding of addition is extremely important, it is expected that you know the basic addition factsup to 10. If you need further practice on these try these hool.com/Games/Addition3.htmlExamples: Addition of Non-negative IntegersExample 3Add. 8 7 Solution: 8 7 15Practice 4Add. 6 8 Solution: Click here to check your answer.It is also important to be able to add larger numbers such as 394 78. In this case we do not want to have to countboxes so a process becomes important. The first thing is that you are careful to add the correct places with each other. Thatis, we must consider place value when adding. Recall the place values listed below.million: hundred-thousand: ten-thousand: thousand: hundred: ten: one: . : tenths: hundredthsTherefore, 1, 234, 567 is read one million, two hundred thirty-four thousand, five hundred sixty-seven. Considering ourproblem 394 78, 3 is in the hundreds column, 9 and 7 are in the tens column and 4 and 8 are in the ones column. Beginningin the ones column 4 8 12 ones. Since we have 12 in the ones column, that is 1 ten and 2 ones, we add the one ten to the9 and the 7 in the tens column. This gives us 17 tens. Again, we must add the 1 hundred in with the 3 hundred so 1 3 4hundred. Giving an answer 394 78 472. As you can see this manner of thinking is not efficient. Typically, we line thecolumns up vertically.11394 784728

Notice that we place the 1’s above the appropriate column.Examples: Vertical AdditionExample 5Solution:Add 8455 97118455 978552Practice 6Add 42, 062 391Solution: Click here to check your answer.Example 7Add 13.45 0.892Solution: In this problem we have decimals but it is worked the same as integer problems by adding the same units. It is oftenhelpful to add in 0 which hold the place value without changing the value of the number. That is, 13.45 0.892 13.450 0.8921113.450 0.89214.342Practice 8Add 321.4 81.732Solution: Click here to check your answer.When we include all integers we must consider problems such as 3 2. We will initially consider the person climbingthe stairs. Once again the person begins at ground level, 0. Negative three would indicate 3 steps down while 2 wouldindicate moving up two steps. As seen below, our stick person ends up one step below ground level which would correspondto 1. So 3 2 1.Next consider the boxes when adding 5 ( 3). In order to view this you must think of black boxes being a negativeand red boxes being a positive. If you match a black box and a red box they neutralize to make 0. That is, 3 red boxesneutralize the 3 black boxes leaving 2 red boxes which means 5 ( 3) 2.9

Consider 2 ( 6). This would be a set of 2 black boxes and 6 black boxes. There are no red boxes to neutralize sothere are a total of 8 black boxes. So, 2 ( 6) 8.For further consideration of this go tohttp://nlvm.usu.edu/en/nav/frames asid 161 g 2 t 1.htmlhttp://www.aaastudy.com/add65 x2.htm#section2As before having to match up boxes or think about climbing up and downstairs can be time consuming so a set ofrules can be helpful for adding 50 27. A generalization of what is occurring depends on the signs of the addends (thenumbers being added). When the addends have different signs you subtract their absolute values. This gives you the numberof “un-neutralized” boxes. The only thing left is to determine whether you have black or red boxes left. This is known byseeing which color of box had more when you started. In 50 27, the addends 50 and 27 have opposite signs so wesubtract their absolute values 50 27 50 27 23. But, since 50 has a larger absolute value than 27 the sum (thesolution to an addition problem) will be negative 23, that is, 50 27 23.In the case when you have the same signs 20 ( 11) or 14 2 we only have the same color boxes so there are no boxesto neutralize each other. Therefore, we just count how many we have altogether (add their absolute values) and denote theproper sign. For 20 ( 11) we have 20 black boxes and 11 black boxes for a total of 31 black boxes so 20 ( 11) 31.Similarly, 14 2 we have 14 red boxes and 2 red boxes for a total of 16 red boxes giving a solution of 14 2 16. A summaryof this discussion is given below.Adding Integers1. Identify the addends.(a) For the same sign:i. Add the absolute value of the addends (ignore the signs)ii. Attach the common sign to your answer(b) For different signs:i. Subtract the absolute value of the addends (ignore the signs)ii. Attach the sign of the addend with the larger absolute valueExamples: AdditionExample 9Solution: 140 90Identify the addends 140 and 90Same sign or differentdifferent signsSubtract the absolute values140 90 50The largest absolute value 140 has the largest absolute valueAttach the sign of addend with the largest absolute value 140 90 50Practice 10 12 4Solution: Click here to check your answer.10

Example 11Solution: 34 ( 55)Identify the addendsSame sign or different?Add the absolute valuesAttach the common sign of addends 34 and 55same signs34 55 89 34 ( 55) 89Practice 12 52 ( 60)Solution: Click here to check your answer.For more practice on addition of integers, click here.Example 13Solution: 1.54 ( 3.2)Identify the addendsSame sign or different?Add the absolute valuesAttach the common sign of addends 1.54 and 3.2same signs1.54 3.2 4.74 1.54 ( 3.2) 4.74Practice 14 20 ( 25.4)Solution: Click here to check your answer.Click here for more practice on decimal addition.Example 15Solution: 8 5 Since there is more than one number insidethe absolute value we must add first 8 5Identify the addends 8 and 5The largest absolute value 8 has the largest absolute valueSame sign or differentdifferent signsSubtract the absolute values8 5 3Attach the sign of addend with the largest absolute value 8 5 3Now take the absolute value 8 5 3 3Practice 16 22 ( 17) Solution: Click here to check your answer.Notice that the absolute value only acts on a single number. You must do the arithmetic inside first.Solutions to Practice Problems:Practice 2 11 11 since 11 is 11 units from 0 (counting the units in red on the number line).Back to TextPractice 46 8 14 Back to TextPractice 611

142062 39142453Back to TextPractice 81 1321.400 81.732403.132Back to TextPractice 10 12 4 8 since 4 red neutralize 4 black boxes leaving 8 black boxes. Back to TextPractice 12 52 ( 60) 112 since the addends are the same we add 52 60 112 and both signs are negative which makes thesolution negative. Back to TextPractice 14 20 ( 25.4) 45.4 since the signs are the same so we add and attach the common sign. Back to TextPractice 16Solution: 22 ( 17) Determining the value of 22 17 first, note that the numbers have the same signs so we addtheir absolute values 22 17 39 and attach the common sign 39. Therefore, 22 ( 17) 39 39 when we takethe absolute value. Back to Text1.2.1Exercises 1.2Evaluate Click here to see examples.1. 50 4. 3.5 2. 33 5. 21 3. 37 6. 55 Add. Click here to see examples.7. 13 510. 36 8913. 167 ( 755)16. 39 ( 29)19. 12 ( 20) 22. 253.2 ( 9.27) 8. 3 1011. 104 199914. 382 ( 675)17. 8 1520. 33 ( 29) 23. 509 3197 9. 59 8812. 2357 54915. 22 ( 20)18. 7 1221. 12.58 ( 78.8) 24. 488 7923 State whether the following are true: Click here to see examples.25. 5 4 5 ( 4) 26. 3 ( 2) 3 2 28. 200 4 200 ( 4) 29. 100 3.1 100 ( 3.1) 27. 12 15 15 12 30. 2 10 2 ( 10) Click here to see the solutions.Click here for more addition practice.1.3SubtractionLet us begin with a simple example of 3 2. Using the stairs application as in addition we would read this as “walk threesteps up then down two steps”.12

We must be able to extend this idea to larger numbers. Consider 1978 322. Just as in addition we must be carefulto line up place values always taking away the smaller absolute value. Again, a vertical subtraction is a good way to keepdigits lined up.1978 3221656Consider 1321 567. When we line this up according to place values we see that we would like to take 7 away from1 in the ones place. This cannot happen. Therefore, we need to borrow from the next column to the left, the tens. As inmoney, 1 ten-dollar bill is worth 10 one-dollar bills so it is that borrowing 1 ten equals 10 ones. We continue borrowing whennecessary as seen below.1 1 113 62 615 6 74 2 11 111 63 62 61 5 6754 0 12 11 1161 63 62 61 567754Examples: Vertical SubtractionExample 113200 4154Solution: Notice that we have to borrow from 2 digits since there was a zero in the column from which we needed to borrow.1 1 93 62 604 1 50 4106046 0 131 63 491 962 601 50 4106046Practice 24501 1728Solution: Click here to check your answer.Example 383.05 2.121Solution: Decimal subtraction is handled the same way as integer subtraction by lining up place values. We also add in extra zeros without changing the value as we did in addition to help us in the subtraction. That is, 83.05 2.121 83.050 2.121.210 4 108 63 . 60 65 60 2 . 1 2 18 0 . 9 2 9Practice 476.4 2.56Solution: Click here to check your answer.13

For a “nice” problem where the minuend (the first number in a subtraction problem) is greater than the subtrahend(the second number in a subtraction problem) we can use the rules we have been discussing. However, we need to know howto handle problems like 3 5. This would read “walk up three steps then down 5 steps” which implies that you are goingbelow ground level leaving you on step 2.Now consider taking away boxes to comprehend the problem 10 4. Using words with the box application this wouldread “ ten red (positive) boxes take away 4 red boxes”. We can see there are 6 red boxes remaining so that 10 4 6.It is possible to use boxes when considering harder problems but a key thing that must be remembered is that a redand black box neutralize each other so it is as if we are adding nothing into our picture. Mathematically, it is as if we areadding zero, since adding zero to any number simply results in the number, (i.e., 5 0 5). So, we can add as many pairs ofred and black boxes without changing the problem. Consider the problem 4 7. We need to add in enough pairs to remove7 red boxes.We see we are left with 3 black boxes so 4 7 3.Examples: Subtraction of IntegersExample 5Solution: 4 514

Therefore, 4 5 9Practice 6 3 7Solution: Click here to check your answer.Example 7Solution: 4 ( 6)Therefore, 4 ( 6) 2Practice 83 ( 2)Solution: Click here to check your answer.Compare Example 7, 4 ( 6), with the problem 4 6.We see that both 4 ( 6) and 4 6 have a solution of 2. Notice that the first number 4 is left alone, we switched thesubtraction to an addition and changed the sign of the second number, 6 to 6. Do you think this will always hold true?The answer is yes.In the case above, we saw subtracting 6 is the same as adding 6. Let us consider another example. Is subtracting 3the same as adding 3? Consider the picture below.As you can see both sides end up with the same result. Although this does not prove “adding the opposite” always works,it does allow us to get an understanding concerning how this works so that we can generalize some rules for subtraction ofintegers.Subtraction 1. Identify the two numbers being subtracted2. Leave the first number alone and add the opposite of the second number(If the second number was positive it should be negative. If it was negativeit should be positive.)3. Follow the rules of addition.15

Examples: SubtractionExample 9Solution: 21 13First number is left alone add the oppositeIdentify the addendsSame sign or differentAdd the absolute valuesAttach the sign 21 ( 13) 21 and 13same signs21 13 34 21 13 21 ( 13) 34Practice 10 11 22Solution: Click here to check your answer.Example 11Solution: 1603 ( 128) 1603 128 1603 and 128different signs 1603 has the largest absolute valueFirst number is left alone add the oppositeIdentify the addendsSame sign or differentThe largest absolute valueSubtract the absolute values Be careful tosubtract the smaller absolute value from the largerAttach the sign of addend with the largest absolute value5 9 131 66 60 63 1 2 81 4 7 5 1603 128 1475Practice 12 201 ( 454)Solution: Click here to check your answer.Example 13Solution:34 543First number is left alone add the oppositeIdentify the addendsSame sign or differentThe largest absolute valueSubtract the absolute values Be careful to34 ( 543)34 and 543different signs 543 has the largest absolute valuesubtract the smaller absolute value from the largerAttach the sign of addend with the largest absolute valuePractice 14 41 77Solution: Click here to check your answer.Example 15Solution:311 ( 729)163 135 64 63 3 45 0 934 543 34 ( 543) 509

First number is left alone add the oppositeIdentify the addendsSame sign or differentAdd the absolute valuesAttach the sign311 729311 and 729same signs1311 7291040311 ( 729) 311 729 1040Practice 16188 560Solution: Click here to check your answer.Example 17Solution:21.3 68.9First number is left alone add the oppositeIdentify the addendsSame sign or different?Subtract the absolute valuesAttach the sign21.3 ( 68.9)21.3 and 68.9different signs68.9 21.347.621.3 68.9 21.3 ( 68.9) 47.6Practice 1815.4 ( 2.34)Solution: Click here to check your answer.Eventually it will be critical that you become proficient with subtraction and no longer need to change the subtractionsign to addition. The idea to keep in mind is that the subtraction sign attaches itself to the number to the right. For example,4 7 3 since we are really looking at 4 ( 7).Try these problems without changing the subtraction over to addition.1. 5 92. 9 73. 10 ( 6)4. 8 ( 7)Click here for answersMore pr