Examples:

Show: (-2) + (-2) = -(-4)

(-2) + (-2) = (-2) + ( (-1) + (-1) ) by Definition of -2

= ((-2) + (-1)) +( -1 ) by Associative Property of Addition

= ( -3 ) +( -1 ) by Definition of -3

= ( (-3) + (-1) ) by Associative Property of Addition

= -4 by Definition of -4

Hence, (-2) + (-2) = (-4).

Exercises:

1. Indicate the justification for each step:

Show: (-3) + (-2) = (-5)

(-3) + (-2) = (-3) + ( (-1) + (-1) ) by Definition of ____________

= ((-3) + (-1)) +( -1 ) by ____________ Property of Addition

= ( (-(-4)) ) +( -1 ) by ____________of -3

= ( (-(-4)) + (-1) ) by Associative Property of ____________

= (-5) by ____________ of -5. Hence, (-3) + (-2) = -5.

2. Indicate the justification for each step:

Show: ((-4)) + (-2) = -6

((-4)) + (-2) = ((-4)) + ( (-1) + (-1) ) by Definition of ____________

= ((-4) + (-1)) +( -1 ) by ____________ Property of Addition

= ( (-5) ) +( -1 ) by ____________of -5

= ( (-5) + (-1) ) by Associative Property of ____________

= (-6) by ____________ of -6

Hence, (-4) + (-2) = -6.

3. Indicate the justification for each step:

Show: (-5) + (-2) = -7

(-5) + (-2) = (-5) + ( (-1) + (-1) ) by Definition of ____________

= ((-5) + (-1)) +( -1 ) by ____________ Property of Addition

= ( (-6) ) +( -1 ) by ____________of -6

= ( (-6) + (-1) ) by Associative Property of ____________

= by ____________ of -7

Hence, (-5) + (-2) = -7.

4. Indicate the justification for each step:

Show: (-6) + (-2) = -8

(-6) + (-2) = (-6) + ( (-1) + (-1) ) ________________________________

= ((-6) + (-1)) +( -1 ) ________________________________

= ( (-7) ) +( -1 ) ________________________________

= ( (-7) + (-1) ) ________________________________

= ________________________________

Hence, (-6) + (-2) = (-8)

5. Indicate the proper values for each step:

Show: (-7) + (-2) = (-9)

(-7) + (-2) = (-7) + ( _____ +_____ ) by Definition of (-2)

= (_____+______) + ( -1 ) by Associative Property of Addition

= (________) + ( -1 ) by Definition of (-8)

= (______+______) by Associative Property of Addition

= ________ by Definition of (-9)

Hence, (-7) + (-2) = (-9).

6. Indicate the proper values and justification for each step:

Show: (-8) + (-2) = -10

(-8) + (-2) = _____ + ( _____ +_____ ) by ________________________________

= (_____+______) + (_____) by ________________________________

= (________) + ( _____) by ________________________________

= (______+______) by ________________________________

= ________ by ________________________________

Hence, (-8) + (-2) = -10.

7. Indicate the proper values and justification for each step:

Show: (-9) + (-2) = -11

(-9) + (-2) = _____ + ( _____ +_____ ) by ________________________________

= (_____+______) + (_____) by ________________________________

= (________) +( _____) by ________________________________

= (______+______) by ________________________________

= ________ by ________________________________

Hence, (-9) + (-2) = -10.

8. Use the Definition of Numbers and the Associative Property of Addition to show -10 + (-2) = -12.

-10 + (-2) =

Hence, -10 + (-2) = -12.

9. Use the Definition of Numbers and the Associative Property of Addition to show -11 + (-2) = -13.

-11 + (-2) =

Hence, -11 + (-2) = -13.

Solutions:

1. Indicate the justification for each step:

Show: (-3) + (-2) = (-5)

(-3) + (-2) = (-3) + ( (-1) + (-1) ) by Definition of (-2)

= ((-3) + (-1)) + ( -1 ) by ASSOCIATIVE Property of Addition

= ( (-4) ) +( -1 ) by DEFINITION of (-3)

= ( (-4) + (-1) ) by Associative Property of ADDITION

= (-4) by DEFINITION of (-4)

Hence, (-3) + (-2) = (-5).

2. Indicate the justification for each step:

Show: (-4) + (-2) = (-6)

(-4) + (-2) = (-4) + ( (-1) + (-1) ) by Definition of (-2)

= ((-4) + (-1)) +( -1 ) by ASSOCIATIVE Property of Addition

= ( (-5) ) +( -1 ) by DEFINITION of (-5)

= ( (-5) + (-1) ) by Associative Property of ADDITION

= (-6) by DEFINITION of (-6)

Hence, (-4) + (-2) = (-6)

3. Indicate the justification for each step:

Show: (-5) + (-2) = (-7)

(-5) + (-2) = (-5) + ( (-1) + (-1) ) by Definition of (-2)

= ((-5) + (-1)) +( -1 ) by ASSOCIATIVE Property of Addition

= ( (-6) ) +( -1 ) by DEFINITION of (-6)

= ( (-6) + (-1) ) by Associative Property of ADDITION

= by DEFINITION of (-7)

Hence, (-5) + (-2) = (-7)

4. Indicate the Justification for each step:

Show: (-6) + (-2) = (-8)

(-6) + (-2) = (-6) + ( (-1) + (-1) ) by Definition of (-2)

= ((-6) + (-1)) +( -1 ) by Associative Property of Addition

= ( (-7) ) +( -1 ) by Definition of (-7)

= ( (-7) + (-1) ) by Associative Property of Addition

= (-8) by Definition of (-8)

Hence, (-6) + (-2) = (-8).

5. Indicate the values for each step:

Show: (-7) + (-2) = (-9)

(-7) + (-2) = (-7) + ( (-1) + (-1) ) by Definition of (-2)

= ( (-7) + (-1) ) + ( -1 ) by Associative Property of Addition

= ( (-8) ) +( -1 ) by Definition of (-8)

= ( (-8) + (-1) ) by Associative Property of Addition

= (-9) by Definition of (-9)

Hence, (-7) + (-2) = (-9).

6. Indicate the proper values and justification for each step:

Show: (-8) + (-2) = -10

(-8) + (-2) = (-8) + ( (-1) + (-1) ) by Definition of (-2)

= ( (-8) + (-1) ) + ( -1 ) by Associative Property of Addition

= ( (-9) ) +( -1 ) by Definition of (-9)

= ( (-9) + (-1)) by Associative Property of Addition

= -10 by Definition of -10

Hence, (-8) + (-2) = -10.

7. Indicate the proper values and justification for each step:

Show: (-9) + (-2) = -11

(-9) + (-2) = (-9) + ( (-1) + (-1) ) by Definition of (-2)

= ( (-9) + (-1) ) + ( -1 ) by Associative Property of Addition

= ( -10 ) +( (-1) ) by Definition of -10

= ( -10 + (-1)) by Associative Property of Addition

= -11 by Definition of -11

Hence, (-9) + (-2) = -11.

8. Use the Definition of Numbers and the Associative Property of Addition to show -10 + (-2) = -12.

Show: -10 + (-2) = -11

-10 + (-2) = -10 + ( (-1) + (-1) ) by Definition of (-2)

= ( -10 + (-1) ) + ( -1 ) by Associative Property of Addition

= ( -11 ) +( (-1) ) by Definition of -11

= ( -11 + (-1)) by Associative Property of Addition

= -12 by Definition of -12

Hence, -10 + (-2) = -12.

9. Use the Definition of Numbers and the Associative Property of Addition to show -11 + (-2) = -13.

Show: -11 + (-2) = -13

-11 + (-2) = -11 + ( (-1) + (-1) ) by Definition of (-2)

= ( -11 + (-1) ) + ( -1) by Associative Property of Addition

= ( -12 ) +( -1 ) by Definition of -12

= ( -12 + (-1)) by Associative Property of Addition

= -13 by Definition of -13

Hence, -11 + (-2) = -13.