Algebra Problems… Solutions Algebra Problems… Solutions © 2007 Herbert I. Gross Set 3 By Herb I. Gross and Richard A. Medeiros next

What signed number is named by: ? Problem #1(a) © 2007 Herbert I. Gross Answer: - 16 next

Answer: - 16 Solution: Since - 9 and - 7 have the same sign we add their magnitudes (7 and 9) and keep the common sign. Hence the answer is -16. next © 2007 Herbert I. Gross

In a signed number the magnitude is the adjective and the sign is the noun. In this case 7 and 9 are adjectives modifying ”negative”. Hence, we add the adjectives (7 + 9) and keep the common noun (negative). So in the same way that… 9 positive + 7 positive = 16 positive, 9 negative + 7 negative = 16 negative. next Note #1a © 2007 Herbert I. Gross next In symbols… = 16 and = - 16

Using a physical model sometimes can make it easier to visualize the answer. next Note #1a © 2007 Herbert I. Gross For example, in terms of profit and loss we know that a $9 loss followed by a $7 loss is a net loss of $16. next

Another Example: next Note #1a © 2007 Herbert I. Gross Suppose during the first hour the temperature decreased by 9 degrees, and during the next hour it decreased by another 7 degrees; then the net decrease in temperature during the two hours was 16 degrees.

next What signed number is named by: ? Problem #1(b) © 2007 Herbert I. Gross Answer: - 2 next

Answer: - 2 Solution: The rule is that when we add two numbers that have different signs we obtain the magnitude of the sum by subtracting the lesser magnitude (in this case, 7) from the greater magnitude (in this case, 9). And the answer will have the sign of the number with the greater magnitude (in this case - 9). In short the answer is given by – (9 – 7)= - 2. next © 2007 Herbert I. Gross

In terms of our adjective/noun theme in this problem 9 and 7 are modifying different nouns (9 is modifying negative and 7 is modifying positive). This explains why we can't add 9 and 7 to get the answer to this problem. next Note #1b © 2007 Herbert I. Gross

To get the answer in a mathematically precise way, we make use of the fact that = 0, and rewrite - 9 as Then… next Note #1b © 2007 Herbert I. Gross = ( ) = = ( ) next

A more “traditional” version of our previous demonstration is to start with… next Note #1b © 2007 Herbert I. Gross = = = Then add + 7 to both sides to obtain… next

In terms of our profit and loss model we are simply saying that the net result of a $9 loss and a $7 profit is a $2 loss. next Profit & Loss © 2007 Herbert I. Gross ProfitLoss $9$9 $7$7 $2$2Net next

In terms of the number line we are saying that the net result of moving 9 units to the left followed by a movement of 7 units to the right is a movement of 2 units to the left. Using 0 as our starting point, this is illustrated below… next Number Line © 2007 Herbert I. Gross next start -2-2 next

If we begin at any temperature and the temperature decreases by 9 degrees, then increases by 7 degrees, the net result is a decrease of 2 degrees in temperature. In our diagram we will start at + 4 degrees. next Temperature Scale © 2007 Herbert I. Gross next starting point ending point

next What signed number is named by: ? Problem #2 © 2007 Herbert I. Gross Answer: - 4 next

Technically speaking our rule for adding signed numbers was given only for the sum of two numbers. However, our rules for arithmetic tell us that, using the associative and commutative properties of addition, we can rearrange and regroup the terms in any way we wish. © 2007 Herbert I. Gross Discussion

Answer: - 4 Solution: One way is to group the positive and the negative terms separately. Thus we may rewrite the problem in the equivalent form… ( ) + ( ) next © 2007 Herbert I. Gross - 17 And by our rule for addition, the above may be rewritten as next

The above solution shows only one of several possible methods. next Discussion © 2007 Herbert I. Gross For example, another way to compute , is to begin with the first two terms This yields + 5, to which we add the third term, + 5 to obtain Finally, we add the fourth term - 14 to + 10 to obtain - 4. next

More visually…. next © 2007 Herbert I. Gross next ( ) ( ) + 10 ( ) = -4-4

may be easier to visualize in terms of profit and loss. For example: next Profit & Loss © 2007 Herbert I. Gross next ProfitLoss $3$3 $8$8 Net $14 $4 $5 $5$5 $10 next $4$4

Tell whether the following statement is true or whether it's false. Give a reason to support your choice. Problem #3 © 2007 Herbert I. Gross Answer: False. - b is the opposite of b. next For any number b, - b is a negative number.

Answer: False. - b is the opposite of b. Solution: The problem here is that “ - ” has 3 different meanings. It is used to indicate… next © 2007 Herbert I. Gross (1) subtraction (“minus”) (2) the sign of a number (“negative”) (3) the opposite of a number. next (As an example of all three meanings, - 9 – - b, is read as “negative 9 minus the opposite of b”.)

As an example in which - b is positive, suppose b = - 7. Then, - b = - ( - 7); but the opposite of - 7 is + 7. In other words, in this illustration - b is a positive number. next Discussion © 2007 Herbert I. Gross Since we have no information about the sign of b, and since no subtraction is indicated, the only alternative is that - b means the opposite of b. next

In algebra, a letter such as b can be used to denote any number, positive or negative. In either case, - b has the opposite sign of b. Thus, whenever b denotes a negative number, - b will denote a positive number. next Application to Algebra © 2007 Herbert I. Gross

next Which number is greater? | + 9| or | - 9| Problem #4 (a) © 2007 Herbert I. Gross Answer: They are equal. next

Answer: They are equal. Solution: Recall that |b| is read as the magnitude of b or as the absolute value of b. The magnitude can never be less than 0. Whether a number is positive or negative depends only on its sign; never on its magnitude. Both + 9 and - 9 have 9 as their magnitude. That is, the meaning of 9 is the same whether we are talking about 9 more than 0 or 9 less than 0. next © 2007 Herbert I. Gross

This is analogous to saying that even though a blue shirt doesn't look like a blue pencil, the word “blue” means the same thing in both cases. The potential problem is that people tend to interpret the phrases “blue shirt” and “blue pencil” as one word. next Meaning & Usage © 2007 Herbert I. Gross

As an historical example, notice that a collection of cattle is called a “herd” but a collection of sheep is called a “flock”. One possible explanation is that different words were chosen because cattle and sheep are different animals. next Historical Example © 2007 Herbert I. Gross

In terms of our profit and loss model whether we're talking about a $9 profit or a $9 loss, the amount of the transaction is $9. The sign simply tells us whether we have a profit of $9 or a loss of $9. next Profit & Loss © 2007 Herbert I. Gross ProfitLoss $9$9 $9$9Net ProfitLoss $9$9 $9$9Net next

Whether we move - 9 or + 9 along the number line, the distance we moved was 9 units regardless of whether it was to the left or to the right. It is the sign that tells us the direction in which we moved. next Number Line © 2007 Herbert I. Gross 9 units next

Which number is greater? | | or | + 13| + | - 9| Problem #4 (b) © 2007 Herbert I. Gross Answer: | + 13| + | - 9| next

Answer: | + 13| + | - 9| Solution: The absolute value symbols act as grouping symbols. Thus | | means that we first add + 13 and - 9 to obtain + 4 as the sum. In other words: | | = | + 4| = 4. next © 2007 Herbert I. Gross On the other hand, | + 13| + | - 9| tells us to first take the absolute values of + 13 and - 9 and then add the results… | + 13| = 13 and | - 9| = 9, so | + 13| + | - 9| = 22 next

This exercise reminds us that the absolute value of a sum is not necessarily the same as the sum of the absolute values. next Reminder © 2007 Herbert I. Gross That is, we showed that… | | ≠ | + 13| + | - 9| next

In fact by taking the absolute values first, we are sure that all the numbers we add are positive. That is, we are adding the magnitudes of the numbers. On the other hand if the two numbers have opposite signs and we add them first, then we are subtracting the lesser magnitude from the greater magnitude. next Note #4b © 2007 Herbert I. Gross

To visualize | |, if we travel 13 units in the positive direction and then 9 units in the negative direction, we are four units to the right of our starting point. next Number Line Interpretation © 2007 Herbert I. Gross To visualize | + 13| + | - 9|, the total distance traveled is | + 13| (that is 13 units in the positive direction) and | - 9| (that is 9 units in the negative direction) or a total of 22 units. next

For example, if you drove your car 13 miles in the east (positive) direction and then drove another 9 miles in the west (negative) direction, you would be 4 miles east of your starting point, but you would have driven a total distance of 22 miles. In science we usually refer to the net change in position as displacement. next Note #4b © 2007 Herbert I. Gross

So if you drove in a circle and ended at the same point at which you started, your displacement would be 0, but the distance you drove would be the circumference of the circle. next Note © 2007 Herbert I. Gross

In summary the absolute value of the sum of two numbers is either equal to or less than the sum of the absolute values of the two numbers. In symbols, if a and b represent the two numbers, then… next Algebra © 2007 Herbert I. Gross |a + b| ≤ |a| + |b| Equality applies if and only if, both numbers have the same sign. next

In terms of positive meaning increasing and negative meaning decreasing, explain the flaw in the following argument... Problem #5 © 2007 Herbert I. Gross

next It’s positive if a person enters the house and negative if a person leaves the house. I know that = 0. Therefore if 2 men enter the house and 3 men leave the house, and I go in, the house will be empty. Problem #5 © 2007 Herbert I. Gross Answer: 0 means no change rather than empty. next

Answer: 0 means no change rather than empty. Discussion: In this problem it should be clear that the signed numbers do not represent the number of people in the house. Rather they represent the change in the number of people in the house. next © 2007 Herbert I. Gross

Geometrically, on the number line we can view signed numbers either as points or as lengths. Physically, this means that a signed number can represent either an amount or a change in an amount. next © 2007 Herbert I. Gross Discussion Continued

In terms of profit and loss = 0 means that a $2 profit followed by a $3 loss and a $1 profit results in no net change. It doesn't mean that there is no money left. next Profit & Loss © 2007 Herbert I. Gross

Looking at signed numbers in terms of directed distance; it means that if we move 2 units to the right, 3 units to the left, and then 1 unit to the right, we are back to the point at which we started. Although we have moved, there is no change in our position. (In scientific terms, our displacement is 0.) next Number Line © 2007 Herbert I. Gross