Mathematics 90 (3122) CA (Class Addendum) 3: Inverse Properties Mt. San Jacinto College Menifee Valley Campus Fall 2009

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1 Mathematics 90 (322) CA (Class Addendum) 3: Inverse Properties Mt. San Jacinto College Menifee Valley Campus Fall 2009 Solutions Name This class handout is worth a maimum of five (5) points. It is due no later than the end of class on Friday 25 September. NOTE: You may need to study this entire handout carefully several times before you begin the eercises it contains. You may need to study eample eercises carefully several times before you attempt the eercise sets that follow them. Also the order in which the eercises occur may not necessarily be the order in which you complete them. If you find that the solutions to a particular eercise set elude you skip to another one. You are being given two weeks to complete this handout because you ll probably need to study it attempt some of the eercises and then take a break continuing with it a day or two later. Every real number has an additive inverse. That is every number has an opposite in the sense that whenever the number and its opposite are added the result is zero. For eample 3 and -3 are additive inverses of one another since 3 + (-3) = 0. Similarly -7 and 7 are opposites of one another since = 0. Zero is its own opposite: = 0. Each variable epression has its own opposite. For eample 8v and -8v are additive 2 2 inverses as are 4z and 4 z. The following property makes the notion of additive inverse precise: The Inverse Property of Addition If is a real number then + (-) = 0 () The following three equations are epressions of The Inverse Property of Addition: = 0 6a + (-6a) = 0 (5 4y) + [- (5 4y)] = 0. Every nonzero real number has a multiplicative inverse. That is every nonzero number has a reciprocal in the sense that whenever the number and its reciprocal are multiplied the result is one. For eample 6 and the fraction /6 are multiplicative inverses since 6(/6) =. Similarly -/9 and -9 are reciprocals since (-/9)(-9) =. Zero does not have a multiplicative inverse since the epression /0 is undefined.

2 Each nonzero variable epression has its own reciprocal. For eample provided y is not zero 5y and /(5y) are reciprocals. Similarly provided is not zero or z is not zero -3z and /(-3z) are multiplicative inverses. The following property makes precise the notion of multiplicative inverse: The Inverse Property of Multiplication If is a real number and is not zero then 0 (2) The following three equations are epressions of the Inverse Property of Multiplication: 3 = 3-72t = 72t 8 5h [8 5h] = Eample. Write an equation that epresses the Inverse Property of Addition utilizing the real number epression 4d. Solution: We are asked to display that the sum of 4d and its opposite is zero. First we must epress the opposite of 4d. By placing a minus sign in front of 4d we obtain -4d an epression that will always represent the opposite of 4d (for more on this see the discussion in the NOTE below.). We can complete the eercise by following the pattern displayed in formula (). That is we ll substitute 4d for in formula (). Then formula () + (-) = 0 becomes 4d + (-4d) = 0. One answer is: 4d + (-4d) = 0. Another answer is: -4d + (4d) = 0. (NOTE: While not a mathematical proof the following table provides suitable evidence that 4d and -4d are always opposites of one another. Since actual values of both 4d and -4d ultimately depend on the specific value of d itself let s start by choosing some arbitrary values for d. The left-hand column of the table lists five such values:

3 and 0. For each of these values for d the corresponding values of 4d and -4d are given in the center and right-hand columns respectively. d 4d -4d Note that for each real value of d the resulting values for 4d and -4d are always opposites of one another. For eample consider the third row of the table. In the first column we see the value of d is -4. Evaluating 4d when d = -4 (i.e. replacing the d with parentheses and inserting -4) yields 4(-4) = -6. Note that this is the value found in the second column third row. Similarly -4d yields the value -4(-4) = 6 (the value in the third column third row). Since 6 and -6 are opposites so are 4d and -4d! That is when d = -4 the table shows that 4d and -4d are indeed opposite real numbers. The same is true for the other four values of d.) Eample 2. Write an equation that epresses the Inverse Property of Multiplication utilizing the real number epression -5t. Solution: Since -5t must be nonzero to have a multiplicative inverse we will assume that -5t is not zero. Therefore we may assume that t itself is not zero. Since the operation is multiplication we don t want the opposite of -5t. Rather we want the reciprocal of -5t. That is we re looking for the epression that when multiplied to -5t yields the real number one. Constructing a fraction with -5t as the denominator and the number one as the numerator does the trick. That is the reciprocal of -5t is: /(-5t) or -/(5t). Note that these fractions represent real numbers. Since t is not zero by assumption neither is -5t or 5t. Therefore we are not dividing by zero so both fractions represent real quantities. We can complete the eercise by following the pattern displayed in formula (2). We ll replace the in formula (2) with -5t. That is formula (2) 0 One answer is: -5t becomes -5t = t 0 5t = t 0. Another answer is: 5t 5t (-5t) = t 0. 3

4 Eercise. (To receive full credit (one point) you must complete at least three of the following four parts correctly). a. Write an equation that epresses the Inverse Property of Addition utilizing the real number epression 3j. Solution: 3j + (-3j) = 0 (Alternate solution: (-3j) + 3j = 0) b. Write an equation that epresses the Inverse Property of Multiplication utilizing the real number epression -7mn Solution: 7mn m 0 n 0 7mn (Alternate solution: ( 7 mn) m 0 n 0 7mn ) c. Write an equation that epresses the Inverse Property of Addition utilizing the following real number epression: 4 - s. Solution: 4 - s + (-[4 s]) = 0 (Alternate solution: (-[4 s]) + (4 s) = 0) d. Write an equation that epresses the Inverse Property of Multiplication utilizing the 4 following real number epression: 5( c y ). 4 4 Solution: 5( c y ) 0 4 5( c y ) c y 4 4 (Alternate solution: 5( c y ) c y 0 ) 4 5( c y ) There is nothing special about the letter utilized in formulas () and (2) above. In Eercise 2 you will be asked to epress the two inverse properties utilizing a variety of symbols. Here are two eamples. Eample 3. Epress the Inverse Property of Addition using the variable c. Solution: One way to interpret this request is that we are being asked to display the Inverse Property of Addition using c rather than in formula (). Therefore replacing with c in + (-) = 0 we have 4

5 c + (-c) = 0 One solution is: c + (-c) = 0. Another solution is: 0 = c + (-c). Eample 4. Epress the Inverse Property of Multiplication utilizing the symbol. Solution: As in Eample 3 we can interpret this request as a restatement of formula (2) using rather than. That is with substituted for 0 becomes One solution is: 0 0 Another solution is: 0 Eercise 2. (To receive full credit (one point) you must complete at least three of the following four parts correctly). a. Epress the Inverse Property of Addition using the variable w. Solution: w + (-w) = 0 (Alternate solution: -w + w = 0) b. Epress the Inverse Property of Multiplication utilizing the variable p. Solution: p (/p) = p 0 (Alternate solution: /p p = p 0) c. Epress the Inverse Property of Addition using the symbol *. Solution: * + (-*) = 0 (Alternate solution: -* + * = 0) d. Epress the Inverse Property of Multiplication utilizing the 5

6 (/@) 0 (Alternate solution: 0) Notice that Eercise 2 provides four additional ways to epress the inverse properties! That is (a) and (c) are simply restatements of formula () and (b) and (d) are just restatements of formula (2). In Eercise 3 you will be asked to finish applications of one of the two inverse laws. In other words as written each equation will be missing a symbol (or two) that you must insert to create an equation that epresses one of the inverse properties. Here are two eamples. Eample 5. Insert the missing symbol(s) (e.g. parentheses a constant or a variable epression) to create an equation that epresses an inverse property. 0 = + 8 Solution: Since we see addition and the equal sign it appears we are to complete an application of the Inverse Property of Addition. The left side of the equation 0 looks complete in that it could already be one side of an equation that epresses an inverse property (of addition). The right side + 8 appears to be missing something!! That is if we inserted the constant -8 between the equal sign the plus sign we d have the epression The entire equation would then become: This equation takes the form 0 = = a + (-a) where the letter a corresponds to the constant -8 (and -a corresponds to the constant 8). But the equation 0 = a + (-a) is equivalent to formula () with a substituted for ). Therefore inserting a -8 between the equal sign and the plus sign completes an application of the Inverse Property of Addition and thus completes the eercise. Eample 6. Insert the missing symbol(s) (e.g. parentheses a constant or a variable epression) to create an equation that epresses an inverse property. 3z = Solution: Since we see division on one side of the equation and a one alone on the other side it appears we might be able to insert symbols to complete an application of the Inverse Property of Multiplication. Since we have a one and only a one on the right 6

7 hand side of the equation it looks like the right hand side is complete in that it could already be one side of an equation that epresses an inverse property (of multiplication). As written the left hand side doesn t even appear equivalent to the right hand side (unless z = -/3) much less an equation that epresses an inverse property. We need to multiply the fraction /(-3z) by a number so that product equals one. What number is needed?: The reciprocal of /(-3z). That is if we insert the factor -3z on the left hand side we ll be done. We d then have -3z = 3z Notice that this equation now has the form 0 where corresponds to -3z ( and / corresponds to /(-3z)). But the equation / = 0 is formula (2)! Therefore inserting -3z as we did completes an application of the Inverse Property of Multiplication and thus completes the eercise. Eercise 3. Insert the missing symbol(s) (e.g. parentheses a constant or a variable epression) to create an equation that epresses an inverse property. (To receive full credit (one point) you must complete at least three of the following four parts correctly). PLEASE USE A PENCIL OR INK OTHER THAN BLACK! a. d + (-d) = 0 b. = ( 5z ) 5z z 0 c. (2 - f) + [-(2 - f )] = 0 d. v(7 - a) v(7 a) = a 7 v 0 In order to recognize an application of a real number property it is necessary to determine how a mathematical epression say in an eercise corresponds to a variable in the formula for the property. The following eamples utilize the inverse properties to illustrate this correspondence. 7

8 Eample 7. The following equation epresses the Inverse Property of Addition. 5d - g + [- (5d g)] = 0 Comparing it to formula () what epression in the equation corresponds to the variable epression - in formula ()? Solution: The equation above and formula () + (-) = 0 are both epressions of the Inverse Property of Addition. One way to determine the correspondence as requested is to write down both equations in a vertical format as follows: 5d - g + [- (5d g)] = 0 + (- ) = 0 Notice that as we read both equations simultaneously as we would read a book (from left to right) we see that the epression (5d g) in the upper equation corresponds to the variable epression - in the lower equation. Therefore the answer is: -(5d g) corresponds to - (or - corresponds to (5d g)). Eample 8. The following equation epresses The Inverse Property of Multiplication. 3 (3 ) = Comparing it to formula (2) which epression in the equation corresponds to the variable epression in formula (2)? Solution: The equation above and formula (2) are both epressions of the inverse property of multiplication. One way to determine the correspondence requested in Eample 8 is write down both equations in a vertical format as follows: 3 (3 ) = Notice that as we read both equations simultaneously as we would read a book (from left to right) we see that the epression in the upper equation corresponds to the 3 8

9 variable in the lower equation. Reading further we see that the epression 3 in the equation corresponds to in formula (2). Therefore the answer is: 3 corresponds to ( or corresponds to 3 ). NOTE: To successfully complete the eercise that follows it is important to use formulas () and (2) as written. That is even though multiplication and addition are commutative it is important not to change the order of and in formula () and and / in formula (2) when answering the questions in the following eercise. At the same time it is important to understand that in general when it comes to representing opposites with variables either of the opposites can be represented by. Then the other number can (and must always) be represented by (read: the opposite of or negative ). For eample 7 and -7 are opposites. If represents 7 then must represent -7. The correspondence between and -7 feels very natural in the sense that the epressions and -7 are both negative quantities. But could just as well represent the -7. In this case - would represent (positive) 7. YES A VARIABLE EXPRESSION WITH A NEGATIVE SIGN CAN REPRESENT A POSITIVE QUANTITY!! Eercise 4. To receive full credit (two points) you must complete all of the following four parts correctly. If you complete two or three parts correctly you ll earn one point. If you complete less than two parts correctly you ll receive zero points. NOTE: Even though addition and multiplication are commutative you must use the equations in formulas () and (2) eactly as they appear when making the comparisons below. a. The following equation is an epression of the Inverse Property of Addition. -(54 + 9e) + (54 + 9e) = 0 Comparing it to formula () which epression in the equation corresponds to the variable in formula ()? -(54 + 9e) corresponds to the variable in formula (). 9

10 b. The following equation is an epression of the Inverse Property of Multiplication. v t ( v t ) = Comparing it to formula (2) which epression in the equation corresponds to the variable epression in formula (2)? v t or (v t) corresponds to the variable epression / in formula (2). c. The following equation is an epression of the Inverse Property of Addition. (4 - c) + (c - 4) = 0 Comparing it to formula () which epression in the equation corresponds to the variable epression - in formula ()? c 4 or (c 4) corresponds to the variable epression in formula (). 0

11 d. The following equation is an epression of the Inverse Property of Multiplication. (-3u) 3u = Comparing it to formula (2) which epression in the equation corresponds to the variable in formula (2)? -3u or (-3u) corresponds to the variable in formula (2).