U Can: Algebra I For Dummies. Sterling Mary Jane

–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

. Use the commutative property to rearrange the factors and the associative property to group the

and the 4 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.

      Chapter 4

      Making Fractions and Decimals Behave

       In This Chapter

      

Simplifying and changing fractions

      

Making proportions work for you

      

Operating on fractions

      

Linking fractions and decimals

      At one time or another, most math students wish that the world were made up of whole numbers only. But those non-whole numbers called fractions really make the world a wonderful place. (Well, that may be stretching it a bit.) In any case, fractions are here to stay, and this chapter helps you delve into them in all their wondrous workings. Compare developing an appreciation for fractions with watching or playing a sport: If you want to enjoy and appreciate a game, you have to understand the rules. You know that this is true if you watch soccer games. That offside rule is hard to understand at first. But, finally, you figure it out, discover the basics of the game, and love the sport. This chapter gets down to basics with the rules involving fractions so you can “play the game.”

      You may not think that decimals belong in a chapter on fractions, but there’s no better place for them. Decimals are just a shorthand notation for the most favorite fractions. Think about the words that are often used and abbreviated, such as Mister (Mr.), Doctor (Dr.), Tuesday (Tues.), October (Oct.), and so on! Decimals are just fractions with denominators of 10, 100, 1,000, and so on, and they’re abbreviated with periods, or decimal points.

      Understanding fractions, where they come from, and why they look the way they do helps when you’re working with them. A fraction has two parts:

       Remember: The denominator of a fraction, or bottom number, tells you the total number of items. The numerator, or top number, tells you how many of that total (the bottom number) are being considered.

      In all the cases using fractions, the denominator tells you how many equal portions or pieces there are. Without the equal rule, you could get different pieces in various