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Section A.1 Arithmetic with Negative Numbers

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permalinkAdding, subtracting, multiplying, dividing, and raising to powers each have peculiarities when using negative numbers. This section reviews arithmetic with signed (both positive and negative) numbers.

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Figure A.1.0.1. Alternative Video Lesson
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Subsection A.1.1 Signed Numbers

permalinkIs it valid to subtract a large number from a smaller one? It may be hard to imagine what it would mean physically to subtract 8 cars from your garage if you only have 1 car in there in the first place. Nevertheless, mathematics has found a way to give meaning to expressions like 1βˆ’8 using signed numbers.

permalinkIn daily life, the signed numbers we might see most often are temperatures. Most people on Earth use the Celsius scale; if you're not familiar with the Celsius temperature scale, think about these examples:

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Figure A.1.1.2. Number line with interesting Celsius temperatures

permalinkFigure A.1.1.2 uses a number line to illustrate these positive and negative numbers. A number line is a useful device for visualizing how numbers relate to each other and combine with each other. Values to the right of 0 are called positive numbers and values to the left of 0 are called negative numbers.

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Warning A.1.1.3. Subtraction Sign versus Negative Sign.

Unfortunately, the symbol we use for subtraction looks just like the symbol we use for marking a negative number. It will help to identify when a β€œminus” sign means β€œsubtract” or means β€œnegative.” The key is to see if there is a number to its left, not counting anything farther left than an open parenthesis. Here are some examples.

  • βˆ’13 has one negative sign and no subtraction sign.

  • 20βˆ’13 has no negative signs and one subtraction sign.

  • βˆ’20βˆ’13 has a negative sign and then a subtraction sign.

  • (βˆ’20)(βˆ’13) has two negative signs and no subtraction sign.

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Checkpoint A.1.1.4.

Identify β€œminus” signs.

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Subsection A.1.2 Adding

permalinkAn easy way to think about adding two numbers with the same sign is to simply (at first) ignore the signs, and add the numbers as if they were both positive. Then make sure your result is either positive or negative, depending on what the sign was of the two numbers you started with.

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Example A.1.2.5. Add Two Negative Numbers.

If you needed to add βˆ’18 and βˆ’7, note that both are negative. Maybe you have this expression in front of you:

βˆ’18+βˆ’7

but that β€œplus minus” is awkward, and in this book you are more likely to have this expression:

βˆ’18+(βˆ’7)

with extra parentheses. (How many subtraction signs do you see? How many negative signs?)

Since both our terms are negative, we can add 18 and 7 to get 25 and immediately realize that our final result should be negative. So our result is βˆ’25:

βˆ’18+(βˆ’7)=βˆ’25

permalinkThis approach works because adding numbers is like having two people tugging on a rope in one direction or the other, with strength indicated by each number. In Example A.1.2.5 we have two people pulling to the left, one with strength 18, the other with strength 7. Their forces combine to pull left with strength 25, giving us our total of βˆ’25, as illustrated in Figure A.1.2.6.

permalinkIf we are adding two numbers that have opposite signs, then the two people tugging the rope are opposing each other. If either of them is using more strength, then the overall effect will be a net pull in that person's direction. And the overall pull on the rope will be the difference of the two strengths. This is illustrated in Figure A.1.2.7.

permalinka number line with Suriya marked at -18 and Kate marked at -7; two thin arrows extend from 0 pointing left, one to Suriya at -18 and one to Kate at -7; a thicker arrow representing Suriya plus Kate points to the left from 0 to -25
Figure A.1.2.6. Working together
permalinka number line with Suriya marked at -15 and Kate marked at 12; one thin arrow extends from 0 pointing left to Suriya at -15, and another thin arrow extends from 0 pointing right to Kate at 12; a thicker arrow representing Suriya plus Kate points to the left from 0 to -3
Figure A.1.2.7. Working in opposition
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Example A.1.2.8. Adding One Number of Each Sign.

Here are four examples of addition where one number is positive and the other is negative.

  1. βˆ’15+12

    We have one number of each sign, with sizes 15 and 12. Their difference is 3. But of the two numbers, the negative number dominated. So the result from adding these is βˆ’3.

  2. 200+(βˆ’100)

    We have one number of each sign, with sizes 200 and 100. Their difference is 100. But of the two numbers, the positive number dominated. So the result from adding these is 100.

  3. 12.8+(βˆ’20)

    We have one number of each sign, with sizes 12.8 and 20. Their difference is 7.2. But of the two numbers, the negative number dominated. So the result from adding these is βˆ’7.2.

  4. βˆ’87.3+87.3

    We have one number of each sign, both with size 87.3. The opposing forces cancel each other, leaving a result of 0.

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Checkpoint A.1.2.9.
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Subsection A.1.3 Subtracting

permalinkPerhaps you can handle a subtraction such as 18βˆ’5, where a small positive number is subtracted from a larger number. There are other instances of subtraction that might leave you scratching your head. In such situations, we recommend that you view each subtraction as adding the opposite number.

Original Adding the Opposite
Subtracting a larger positive number: 12βˆ’30 12+(βˆ’30)
Subtracting from a negative number: βˆ’8.1βˆ’17 βˆ’8.1+(βˆ’17)
Subtracting a negative number: 42βˆ’(βˆ’23) 42+23

permalinkThe benefit is that perhaps you already mastered addition with positive and negative numbers, and this strategy that you convert subtraction to addition means you don't have all that much more to learn. These examples might be computed as follows:

12βˆ’30=12+(βˆ’30)=βˆ’18
βˆ’8.1βˆ’17=βˆ’8.1+(βˆ’17)=βˆ’25.1
42βˆ’(βˆ’23)=42+23=65
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Checkpoint A.1.3.10.
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Subsection A.1.4 Multiplying

permalinkMaking sense of multiplication of negative numbers isn't quite so straightforward, but it's possible. Should the product of 3 and βˆ’7 be a positive number or a negative number? Remembering that we can view multiplication as repeated addition, we can see this result on a number line:

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Figure A.1.4.11. Viewing 3β‹…(βˆ’7) as repeated addition

permalinkFigure A.1.4.11 illustrates that 3β‹…(βˆ’7)=βˆ’21, and so it would seem that a positive number times a negative number will always give a negative result. (Note that it would not change things if the negative number came first in the product, since the order of multiplication doesn't affect the result.)

permalinkWhat about the product βˆ’3β‹…(βˆ’7), where both factors are negative? Should the product be positive or negative? If 3β‹…(βˆ’7) can be seen as adding βˆ’7 three times as in Figure A.1.4.12, then it isn't too crazy to interpret βˆ’3β‹…(βˆ’7) as subtracting βˆ’7 three times, as in Figure A.1.4.12.

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Figure A.1.4.12. Viewing βˆ’3β‹…(βˆ’7) as repeated subtraction

permalinkThis illustrates that βˆ’3β‹…(βˆ’7)=21, and it would seem that a negative number times a negative number always gives a positive result.

permalinkPositive and negative numbers are not the whole story. The number 0 is neither positive nor negative. What happens with multiplication by 0? You can choose to view 7β‹…0 as adding the number 0 seven times. And you can choose to view 0β‹…7 as adding the number 7 zero times. Either way, you really added nothing at all, which is the same as adding 0.

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Checkpoint A.1.4.14.
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Subsection A.1.5 Powers

permalinkFor early sections of this book the only exponents you will see will be the natural numbers : {1,2,3,…}. But negative numbers can and will arise as the base of a power.

permalinkAn exponent is a shorthand for how many times to multiply by the base. For example,

(βˆ’2)5 means (βˆ’2)β‹…(βˆ’2)β‹…(βˆ’2)β‹…(βˆ’2)β‹…(βˆ’2)⏞5 instances

permalinkWill the result here be positive or negative? Since we can view (βˆ’2)5 as repeated multiplication, and we now understand that multiplying two negatives gives a positive result, this expression can be thought of this way:

(βˆ’2)β‹…(βˆ’2)⏟positiveβ‹…(βˆ’2)β‹…(βˆ’2)⏟positive⏟positiveβ‹…(βˆ’2)

permalinkand that lone last negative number will be responsible for making the final product negative.

permalinkMore generally, if the base of a power is negative, then whether or not the result is positive or negative depends on if the exponent is even or odd. It depends on whether or not the factors can all be paired up to β€œcancel” negative signs, or if there will be a lone factor left by itself.

permalinkOnce you understand whether the result is positive or negative, for a moment you may forget about signs. Continuing the example, you may calculate that 25=32, and then since we know (βˆ’2)5 is negative, you can report

(βˆ’2)5=βˆ’32
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Warning A.1.5.15. Negative Signs and Exponents.

Expressions like βˆ’34 may not mean what you think they mean. What base do you see here? The correct answer is 3. The exponent 4 only applies to the 3, not to βˆ’3. So this expression, βˆ’34, is actually the same as βˆ’(34), which is βˆ’81. Be careful not to treat βˆ’34 as having base βˆ’3. That would make it equivalent to (βˆ’3)4, which is positive 81.

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Checkpoint A.1.5.16.
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Subsection A.1.6 Summary

Addition

Add two negative numbers: add their positive counterparts and make the result negative.

Add a positive with a negative: find their difference using subtraction, and keep the sign of the dominant number.

Subtraction

Any subtraction can be converted to addition of the opposite number. For all but the most basic subtractions, this is a useful strategy.

Multiplication

Multiply two negative numbers: multiply their positive counterparts and make the result positive.

Multiply a positive with a negative: multiply their positive counterparts and make the result negative.

Multiply any number by 0: the result will be 0.

Division (not discussed in this section)

Division by some number is the same as multiplication by its reciprocal. So the multiplication rules can be adopted.

Division of 0 by any nonzero number always results in 0.

Division of any number by 0 is always undefined.

Powers

Raise a negative number to an even power: raise the positive counterpart to that power.

Raise a negative number to an odd power: raise the positive counterpart to that power, then make the result negative.

Expressions like βˆ’24 mean βˆ’(24), not (βˆ’2)4.

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Exercises A.1.7 Exercises

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1.

Add the following.

  1. βˆ’9+(βˆ’3)

  2. βˆ’6+(βˆ’5)

  3. βˆ’1+(βˆ’7)

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2.

Add the following.

  1. βˆ’8+(βˆ’1)

  2. βˆ’5+(βˆ’6)

  3. βˆ’1+(βˆ’9)

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3.

Add the following.

  1. 5+(βˆ’9)

  2. 5+(βˆ’1)

  3. 8+(βˆ’8)

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4.

Add the following.

  1. 5+(βˆ’6)

  2. 8+(βˆ’2)

  3. 8+(βˆ’8)

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5.

Add the following.

  1. βˆ’7+1

  2. βˆ’3+10

  3. βˆ’3+3

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6.

Add the following.

  1. βˆ’9+2

  2. βˆ’4+7

  3. βˆ’3+3

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7.

Add the following.

  1. βˆ’71+(βˆ’16)

  2. βˆ’32+87

  3. 78+(βˆ’23)

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8.

Add the following.

  1. βˆ’61+(βˆ’47)

  2. βˆ’84+62

  3. 77+(βˆ’69)

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9.

Subtract the following.

  1. 3βˆ’7

  2. 8βˆ’2

  3. 5βˆ’19

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10.

Subtract the following.

  1. 4βˆ’10

  2. 5βˆ’1

  3. 5βˆ’14

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11.

Subtract the following.

  1. βˆ’2βˆ’3

  2. βˆ’8βˆ’4

  3. βˆ’5βˆ’5

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12.

Subtract the following.

  1. βˆ’1βˆ’2

  2. βˆ’6βˆ’3

  3. βˆ’5βˆ’5

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13.

Subtract the following.

  1. βˆ’5βˆ’(βˆ’8)

  2. βˆ’6βˆ’(βˆ’1)

  3. βˆ’5βˆ’(βˆ’5)

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14.

Subtract the following.

  1. βˆ’1βˆ’(βˆ’7)

  2. βˆ’9βˆ’(βˆ’2)

  3. βˆ’5βˆ’(βˆ’5)

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15.

Perform the given addition and subtraction.

  1. βˆ’18βˆ’2+(βˆ’9)

  2. 7βˆ’(βˆ’13)+(βˆ’11)

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16.

Perform the given addition and subtraction.

  1. βˆ’17βˆ’9+(βˆ’5)

  2. 4βˆ’(βˆ’14)+(βˆ’17)

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17.

Perform the given addition and subtraction.

  1. βˆ’16βˆ’5+(βˆ’1)

  2. 1βˆ’(βˆ’14)+(βˆ’11)

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18.

Perform the given addition and subtraction.

  1. βˆ’14βˆ’2+(βˆ’7)

  2. 9βˆ’(βˆ’14)+(βˆ’17)

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19.

Multiply the following.

  1. (βˆ’9)β‹…(βˆ’1)

  2. (βˆ’4)β‹…5

  3. 7β‹…(βˆ’2)

  4. (βˆ’6)β‹…0

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20.

Multiply the following.

  1. (βˆ’8)β‹…(βˆ’2)

  2. (βˆ’7)β‹…3

  3. 7β‹…(βˆ’5)

  4. (βˆ’5)β‹…0

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21.

Multiply the following.

  1. (βˆ’1)β‹…(βˆ’6)β‹…(βˆ’3)

  2. 7β‹…(βˆ’7)β‹…(βˆ’1)

  3. (βˆ’86)β‹…(βˆ’78)β‹…0

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22.

Multiply the following.

  1. (βˆ’1)β‹…(βˆ’4)β‹…(βˆ’5)

  2. 6β‹…(βˆ’7)β‹…(βˆ’4)

  3. (βˆ’84)β‹…(βˆ’66)β‹…0

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23.

Multiply the following.

  1. (βˆ’1)(βˆ’2)(βˆ’2)(βˆ’2)

  2. (3)(βˆ’3)(2)(βˆ’3)

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24.

Multiply the following.

  1. (βˆ’1)(βˆ’1)(βˆ’3)(βˆ’1)

  2. (1)(βˆ’3)(βˆ’3)(βˆ’1)

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25.

Evaluate the following.

  1. βˆ’64βˆ’8

  2. 54βˆ’6

  3. βˆ’637

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26.

Evaluate the following.

  1. βˆ’28βˆ’7

  2. 35βˆ’5

  3. βˆ’287

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27.

Evaluate the following.

  1. βˆ’6βˆ’1

  2. 10βˆ’1

  3. 110βˆ’110

  4. βˆ’17βˆ’17

  5. 110

  6. 0βˆ’3

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28.

Evaluate the following.

  1. βˆ’5βˆ’1

  2. 8βˆ’1

  3. 150βˆ’150

  4. βˆ’20βˆ’20

  5. 100

  6. 0βˆ’8

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29.

Evaluate the following.

  1. (βˆ’9)2

  2. βˆ’42

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30.

Evaluate the following.

  1. (βˆ’7)2

  2. βˆ’62

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31.

Evaluate the following.

  1. (βˆ’3)3

  2. βˆ’13

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32.

Evaluate the following.

  1. (βˆ’2)3

  2. βˆ’43

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33.

Evaluate the following.

  1. 42

  2. 23

  3. (βˆ’4)2

  4. (βˆ’3)3

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34.

Evaluate the following.

  1. 52

  2. 43

  3. (βˆ’3)2

  4. (βˆ’5)3

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35.

Evaluate the following.

  1. 110

  2. (βˆ’1)11

  3. (βˆ’1)12

  4. 020

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36.

Evaluate the following.

  1. 15

  2. (βˆ’1)13

  3. (βˆ’1)18

  4. 018

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37.

Simplify without using a calculator.

βˆ’6.53+(βˆ’53.7)

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38.

Simplify without using a calculator.

βˆ’7.17+(βˆ’33.3)

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39.

Simplify without using a calculator.

9.8βˆ’1.93

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40.

Simplify without using a calculator.

9.5βˆ’5.73

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41.

Simplify without using a calculator.

βˆ’4.43+5.2

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42.

Simplify without using a calculator.

βˆ’2.13+6.7

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43.

Simplify without using a calculator.

βˆ’4.83βˆ’(βˆ’6.4)

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44.

Simplify without using a calculator.

βˆ’2.63βˆ’(βˆ’7.1)

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45.

Simplify without using a calculator.

57βˆ’6.33

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46.

Simplify without using a calculator.

63βˆ’1.03

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47.

Simplify without using a calculator.

βˆ’70+4.73

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48.

Simplify without using a calculator.

βˆ’76+8.42

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49.

It’s given that 94β‹…32=3008. Use this fact to calculate the following without using a calculator.

9.4(βˆ’0.032)=

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50.

It’s given that 19β‹…79=1501. Use this fact to calculate the following without using a calculator.

1.9(βˆ’7.9)=

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51.

It’s given that 26β‹…16=416. Use this fact to calculate the following without using a calculator.

(βˆ’2.6)(βˆ’0.016)

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52.

It’s given that 33β‹…54=1782. Use this fact to calculate the following without using a calculator.

(βˆ’3.3)(βˆ’5.4)

Applications
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53.

Consider the following situation in which you borrow money from your cousin:

  • On June 1st, you borrowed 1200 dollars from your cousin.

  • On July 1st, you borrowed 490 more dollars from your cousin.

  • On August 1st, you paid back 690 dollars to your cousin.

  • On September 1st, you borrowed another 820 dollars from your cousin.

How much money do you owe your cousin now?

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54.

Consider the following scenario in which you study your bank account.

  • On Jan. 1, you had a balance of βˆ’310 dollars in your bank account.

  • On Jan. 2, your bank charged 40 dollar overdraft fee.

  • On Jan. 3, you deposited 870 dollars.

  • On Jan. 10, you withdrew 750 dollars.

What is your balance on Jan. 11?

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55.

A mountain is 1100 feet above sea level. A trench is 360 feet below sea level. What is the difference in elevation between the mountain top and the bottom of the trench?

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56.

A mountain is 1200 feet above sea level. A trench is 420 feet below sea level. What is the difference in elevation between the mountain top and the bottom of the trench?

Challenge
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57.

Select the correct word to make each statement true.

  1. A positive number minus a positive number is

    • sometimes

    • always

    • never

    negative.

  2. A negative number plus a negative number is

    • sometimes

    • always

    • never

    negative.

  3. A positive number minus a negative number is

    • sometimes

    • always

    • never

    positive.

  4. A negative number multiplied by a negative number is

    • sometimes

    • always

    • never

    negative.