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In This Chapter

Embracing the different types of grouping symbols

Distributing over addition and subtraction

Incorporating inverses and identities

Utilizing the associative and commutative rule

Algebra has rules for everything, including a sort of shorthand notation to save time and space. The notation that comes with particular properties cuts down on misinterpretation because it’s very specific and universally known. (I give the guidelines for doing operations like addition, subtraction, multiplication, and division in Chapter 2.) In this chapter, you see the specific rules that apply when you use grouping symbols and rearrange terms. You also find how opposites attract – or not – in the form of inverses and identities.

Getting a Grip on Grouping Symbols

The most commonly used grouping symbols in algebra are (in order from most to least common):

✓ Parentheses ( )

✓ Brackets [ ]

✓ Braces { }

✓ Fraction lines /

✓ Radicals

✓ Absolute value symbols | |

Here’s what you need to know about grouping symbols: You must compute whatever is inside them (or under or over, in the case of the fraction line) first, before you can use that result to solve the rest of the problem. If what’s inside isn’t or can’t be simplified into one term, then anything outside the grouping symbol that multiplies one of the terms has to multiply them all – that’s the distributive property, which I cover in the next section.

Examples

Q. 16 – (4 + 2) =

A. Add the 4 and 2; then subtract the result from the 16: 16 – (4 + 2) = 16 – 6 = 10.

Q. Simplify 2[6 – (3 – 7)].

A. First subtract the 7 from the 3; then subtract the –4 from the 6 by changing it to an addition problem. You can then multiply the 2 by the 10: 2[6 – (3 – 7)] = 2[6 – (–4)] = 2[6 + 4] = 2[10] = 20.

Q.

A. Combine what’s in the absolute value and parentheses first, before combining the results:

When you get to the three terms with subtract and add, 1 – 11 + 18, you always perform the operations in order, reading from left to right. (See Chapter 7 for more on this process, called the order of operations.)

Q.

A. You have to complete the work in the denominator first before dividing the 32 by that result:

Practice Questions

1. 3(2 – 5) + 14 =

2. 4[3(6 – 8) + 2(5 + 9)] – 11 =

3. 5{8[2 + (6 – 3)] – 4} =

4.

5.

6.

Practice Answers

1. 5.

2. 77.

3. 180.

4.

5. –56.

6. 8.

Distributing the Wealth

The distributive property is used to perform an operation on each of the terms within a grouping symbol. The following rules show distributing multiplication over addition and distributing multiplication over subtraction:

Examples

Q. 3(6 – 4) =

A. First, distribute the 3 over the 6 – 4: 3(6 – 4) = 3 × 6 – 3 × 4 = 18 – 12 = 6. Another (simpler) way to get the correct answer is just to subtract the 4 from the 6 and then multiply: 3(2) = 6. However, when you can’t or don’t want to combine what’s in the grouping symbols, you use the distributive property.

Q.

A.

Practice Questions

1. 4(7 + y) =

2. –3(x – 11) =

3.

4.

5.

6.

Practice Answers

1.

2.

3.

4. –5.

5.

6.

Making Associations Work

The associative property has to do with how the numbers are grouped when you perform operations on more than two numbers. Think about what the word associate means. When you associate with someone, you’re close to the person, or you’re in the same group with the person. Say that Anika, Becky, and Cora associate. Whether Anika drives over to pick up Becky and the two of them go to Cora’s and pick her up, or Cora is at Becky’s house and Anika picks up both of them at the same time, the same result occurs – the three ladies are all in the car at the end.

Tip: The associative property means that even if the grouping of the operation changes, the result remains the same. (If you need a reminder about grouping, refer to “Getting a Grip on Grouping Symbols,” earlier in this chapter.) Addition and multiplication are associative. Subtraction and division are not associative operations. So,

You can always find a few cases where the associative property works even though it isn’t supposed to. For example, in the subtraction problem 5 – (4 – 0) = (5 – 4) – 0, the property seems to work. Also, in the division problem 6 ÷ (3 ÷ 1) = (6 ÷ 3) ÷ 1, it seems to work. I just picked numbers very carefully that would make it seem like you could associate with subtraction and division. Although there are exceptions, a rule must work all the time, not just in special cases.

Here’s how the associative property works:

✓ 4 + (5 + 8) = 4 + 13 = 17 and (4 + 5) + 8 = 9 + 8 = 17, so 4 + (5 + 8) = (4 + 5) + 8

✓ 3 – (2 × 5) = 3 × 10 = 30 and (3 × 2) × 5 = 6 × 5 = 30, so 3 × (2 × 5) = (3 × 2) × 5

✓

✓ 3.2 × (5 × 4.8) = 3.2 × 24 = 76.8 and (3.2 × 5) × 4.8 = 16 × 4.8 = 76.8

Remember: This rule is special to addition and multiplication. It doesn’t work for subtraction or division. You’re probably wondering why even use this rule? Because it can sometimes make the computation easier.

✓ Instead of doing 5 + (–5 + 17), change it to [5 + (–5)] + 17 = 0 + 17 = 17.

✓ Instead of , do .

Examples

Q. –14 + (14 + 23) =

A. Re-associate the terms and then add the first two together: –14 + (14 + 23) = (–14 + 14) + 23 = 0 + 23 = 23.

Q. 4(5 · 6) =

A. You can either multiply the way the problem is written, 4(5 · 6) = 4(30) = 120, or you can re-associate and multiply the first two factors first: (4 · 5) 6 = (20)6 = 120.

Practice Questions

1. 16 + (–16 + 47) =

2. (5 – 13) + 13 =

3.

4.

Practice Answers

1. 47.

2. 5.

3. 70.

4. 110.

Computing by Commuting

Before discussing the commutative property, take a look at the word commute. You probably commute to work or school and know that whether you’re traveling from home to work or from work to home, the distance is the same: The distance doesn’t change because you change directions (although getting home during rush hour may make that distance seem longer).

The same principle is true of some algebraic operations: It doesn’t matter whether you add 1 + 2 or 2 + 1, the answer is still 3. Likewise, multiplying 2 · 3 or 3 · 2 yields 6.

Tip: The commutative property means that you can change the order of the numbers in an operation without affecting the result. Addition and multiplication are commutative. Subtraction and division are not. So,

In general, subtraction and division are not commutative. The special cases occur when you choose the numbers carefully. For example, if a and b are the same number, then the subtraction appears to be commutative because switching the order doesn’t change the answer. In the case of division, if a and b are opposites, then you get –1 no matter which order you divide them in. By the way, this is why, in mathematics, big deals are made about proofs. A few special cases of something may work, but a real rule or theorem has to work all the time.

Take a look at how the commutative property works:

✓ 4 + 5 = 9 and 5 + 4 = 9, so 4 + 5 = 5 + 4

✓ 3 · (–7) = –21 and (–7) · 3 = –21, so 3 · (–7) = (–7) · 3

✓ 6.3 + 5.7 = 12 and 5.7 + 6.3 = 12, so 6.3 + 5.7 = 5.7 + 6.3

✓

You can use this rule to your advantage when doing math computations. In the following two examples, the associative rule finishes off the problems after changing the order.

Examples

Q.

A. You don’t really want to multiply fractions unless necessary. Notice that the first and last factors are multiplicative inverses of one another: . The second and last factors were reversed.

Q. –3 + 16 + 303 =

A. The second and last terms are reversed, and then the first two terms are grouped.

– 3 + 16 + 303 = –3 + 303 + 16 = (–3 + 303) + 16 = 300 + 16 = 316.

Practice Questions

1. 8 + 5 + (–8) =

2. 5 · 47 · 2 =

3.

4. –23 + 47 + 23 – 47 + 8 =

Practice Answers

1. 5.

2. 470.

3. 78.

4. 8.

Investigating Inverses

In mathematics, the inverse of a number is tied to a specific operation.

The additive inverse of the number 5 is –5; the additive inverse of the number is . When you add a number and its additive inverse together, 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 . 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.