U Can: Algebra I For Dummies. Sterling Mary Jane

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you always get 0, the additive identity. Every real number has an additive inverse, even the number 0. The number 0 is its own additive inverse. And all real numbers (except 0) and their inverses have opposite signs; the number 0 is neither positive nor negative, so there is no sign.

      The multiplicative inverse of the number 5 is

; the multiplicative inverse of the number is –3. When you multiply a number and its multiplicative inverse together, you always get 1, the multiplicative identity. Every real number except the number 0 has a multiplicative inverse. A number and its multiplicative inverse are always the same sign.

       Examples

      Q. Find the additive and multiplicative inverses of the number –14.

      A. The additive inverse is 14, because –14 + 14 = 0. The multiplicative inverse of –14 is

, because .

      Q. Find the additive and multiplicative inverses of the number

.

      A. The additive inverse is

, because . The multiplicative inverse of is , because .

       Practice Questions

      Find the additive and multiplicative inverses of the number given.

      1. 11

      2.

      3.

      4. –1

       Practice Answers

      1. – 11 and

. The sum of 11 and –11 is 0; the product of 11 and is 1.

      2.

and –3. The sum of and is 0; the product of and –3 is 1.

      3.

and . The sum of and is 0; the product of and is 1.

      4. 1 and –1. The sum of –1 and 1 is 0; the product of –1 and –1 is 1. The number is its own multiplicative inverse.

Identifying Identities

      The term identity in mathematics is most frequently used in terms of a specific operation. When using addition, the additive identity is the number 0. You can think of it as allowing another number to keep its identity when 0 is added. If you add 7 + 0, the result is 7. The number 7 doesn’t change. When using multiplication, the multiplicative identity is the number 1. When you multiply 7 × 1, the result is 7. Again, the number 7 doesn’t change.

      When adding a number and its additive inverse together, you get the additive identity. So –5 + 5 = 0. And when multiplying a number and its multiplicative inverse together, you get the multiplicative identity. Multiplying,

.

       Examples

      Q. Use an additive identity to change the expression 4x + 5 to an expression with only the variable term.

      A. The additive inverse of 5 is –5. If you add –5 to the expression, you have 4x + 5 + (–5). Use the associative property to group the 5 and –5 together:

. The sum of a number and its additive inverse is 0, so the expression becomes 4x + 0. Because 0 is the additive identity, 4x + 0 = 4x.

       Practice Questions

      1. Use an additive identity to change the expression 9x – 8 to one with only the variable term.

      2. Use an additive identity to change the expression 6 – 3x to one with only the variable term.

      3. Use a multiplicative identity to change the expression –7x to one with only the variable factor.

      4. Use a multiplicative identity to change the expression

to one with only the variable factor.

       Practice Answers

      1. Use 8. The additive inverse of –8 is 8. If you add 8 to the expression, you have 9x – 8 + 8. Use the associative property to group the –8 and 8 together: 9x + (–8 + 8). The sum of a number and its additive inverse is 0, so the expression becomes 9x + 0. Because 0 is the additive identity, 9x + 0 = 9x.

      2. Use –6. The additive inverse of 6 is –6. If you add –6 to the expression, you have 6 + (–6) – 3x. Use the associative property to group the 6 and –6 together:

. The sum of a number and its additive inverse is 0, so the expression becomes 0 – 3x. Because 0 is the additive identity, 0 – 3x = –3x.

      3. Use

. The multiplicative inverse of –7 is . If you multiply the expression by , you have . Use the commutative property to rearrange the factors and the associative property to group the –7 and together: . The product of a number and its multiplicative inverse is 1, so the expression becomes 1x. Because 1 is the multiplicative identity, 1x = x.

      4. Use 4. The expression

can be written as . The multiplicative inverse of is 4. If you multiply the expression by 4, you have

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