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Topic 2.1.2Topic 2.1.2

Getting Rid of Grouping SymbolsGetting Rid of Grouping Symbols

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Topic2.1.2

Getting Rid of Grouping SymbolsGetting Rid of Grouping Symbols

California Standard:4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x – 5) + 4(x – 2) = 12

What it means for you:You’ll use the distributive property to simplify expressions.

Key words:• distributive property• commutative property

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Topic2.1.2

Getting Rid of Grouping SymbolsGetting Rid of Grouping Symbols

You already saw the distributive property in Topic 1.2.7.

In this Topic you’ll simplify expressions by using the distributive property to get rid of grouping symbols.

For example: 3(b + c) = 3b + 3c

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Topic2.1.2

The Distributive Property Removes Grouping Symbols

Getting Rid of Grouping SymbolsGetting Rid of Grouping Symbols

The expression 5(3x + 2) + 2(2x – 1) can be simplified — both parts have an “x” term and a constant term.

To simplify an expression like this, you first need to get rid of the grouping symbols.

The way to do this is to use the distributive property of multiplication over addition: a(b + c) = ab + ac.

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Topic2.1.2

Example 1

Solution follows…

Getting Rid of Grouping SymbolsGetting Rid of Grouping Symbols

Simplify 5(3x + 2) + 2(2x – 1).

Solution

= 19x + 8

5(3x + 2) + 2(2x – 1)

= 15x + 10 + 4x – 2

= 15x + 4x + 10 – 2

Distributive property

Commutative property of addition

Given expression

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Topic2.1.2

Guided Practice

Solution follows…

Getting Rid of Grouping SymbolsGetting Rid of Grouping Symbols

In Exercises 1–7, simplify the following expressions:

2(4x + 5) + 8

12(5a – 8) + 4x + 3

6(2j + 3c) + 8(5c + 4z)

10(x + 2) + 7(3 – 4x)

6(a – b) + 4(2b – 3)

5(3x + 4) + 3(4x + 10) + 2(8x + 9)

8(2n – 3) + 9(4n – 5) + 4(3n + 7)

1.

2.

3.

4.

5.

6.

7.

8x + 18

60a + 4x – 93

12j + 58c + 32z

–18x + 41

6a + 2b – 12

43x + 68

64n – 41

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Topic2.1.2

Take Care when Multiplying by a Negative Number

Getting Rid of Grouping SymbolsGetting Rid of Grouping Symbols

If a number outside a grouping symbol is negative, like in –7(2x + 1), you have to remember to use the multiplicative property of –1.

This means that the signs of the terms within the grouping symbols will change: “+” signs will change to “–” signs and vice versa.

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Topic2.1.2

Example 2

Solution follows…

Getting Rid of Grouping SymbolsGetting Rid of Grouping Symbols

Simplify the following:

Solutions

The two negative terms inside the grouping symbols are multiplied by the negative term outside. They both become positive.

–3(5x – 4) = –15x + 12

The +2x and +1 become negative.

–7(2x + 1) = –14x – 7

–6(–x – 3) = 6x + 18

a) –7(2x + 1) b) –6(–x – 3) c) –3(5x – 4)

b)

a)

c)

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Topic2.1.2

Example 3

Solution follows…

Getting Rid of Grouping SymbolsGetting Rid of Grouping Symbols

Simplify the expression 4(2x – 1) – 5(x – 2). Show your steps.

Solution

4(2x – 1) – 5(x – 2)

= 8x – 4 – 5x + 10

= 8x – 5x – 4 + 10

= 3x + 6

Given expression

Distributive property

Commutative property of addition

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Topic2.1.2

Guided Practice

Solution follows…

Getting Rid of Grouping SymbolsGetting Rid of Grouping Symbols

In Exercises 8–13, simplify each algebraic expression:

8. –2(5a – 3c)

9. –8(3c – 2)

10. –2(–3x – 4) + 4(6 – 2x)

11. 7(2a + 9) – 4(a + 11)

12. –8(2y + 4) – 5(y + 4)

–10a + 6c

–24c + 16

–2x + 32

10a + 19

–21y – 52

–2n + 813. –2 n + 2 – 3 n – 4 1

2

1

3

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Topic2.1.2

Guided Practice

Solution follows…

Getting Rid of Grouping SymbolsGetting Rid of Grouping Symbols

14. Simplify 12(2n – 7) – 9(3 – 4n) + 6(4x – 9).

15. Simplify 5(x – 2) – 7(–4x + 3) – 3(–2x).

60n + 24x – 165

39x – 31

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Topic2.1.2

Independent Practice

Solution follows…

Getting Rid of Grouping SymbolsGetting Rid of Grouping Symbols

In Exercises 1–7, simplify the algebraic expressions:

1. –4(a + 2b)

2. 3(2n + 4) + n(–4)

3. 5(3b – 2q) – (3q + 4)

4. –9(2 + 3b) + 3(3 – 2b)

5. –17(3a – 5b) – 4(x + 3)

6. –2(10n – 4x) + 2(n + 6x) – 3(7x – 2n)

7. –7(3p – 9q) – 4(25q – 3r) + 12(5p + 8)

–4a – 8b

2n + 12

15b – 13q – 4

–33b – 9

–51a + 85b – 4x – 12

–12n – x

39p – 37q + 12r + 96

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Topic2.1.2

Independent Practice

Solution follows…

Getting Rid of Grouping SymbolsGetting Rid of Grouping Symbols

In Exercises 8–12, find and simplify an expression for the perimeter of the shape shown.

8. 10.

11. 12.

9.

P = 6x + 3 P = 8x – 5 P = 8 – 2x

P = 14n – 6 P = 60 – 18n

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Topic2.1.2

Independent Practice

Solution follows…

Getting Rid of Grouping SymbolsGetting Rid of Grouping Symbols

In Exercises 13–15, simplify the algebraic expressions:

14. –0.1(25n + 46) – 0.8(16n – 5) + 0.4(3 – 4n) –16.9n + 0.6

5

64.15x +

13. (3x + 4) – (4x – 8) 3

8

1

27

8– x +

11

2

15. – (5 – 9x) – (2x – 4) – 0.2(–7x – 10) 1

3

1

8

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Topic2.1.2

Independent Practice

Solution follows…

Getting Rid of Grouping SymbolsGetting Rid of Grouping Symbols

16. Mia has 4(x + 1) dolls, where x is Mia's age. Madeline has 5(2x + 3) dolls.

Write and simplify an algebraic expression showing the total number of dolls in Mia and Madeline's collection.

17. Ruby, Sara, and Keisha are counting stamps. If x represents Ruby's age, she has 4(x – 4) stamps, Sara has 2(8 – x) stamps, and Keisha has 8(7 + 2x) stamps.

Write and simplify an algebraic expression showing the total number of stamps owned by the three friends.

14x + 19

18x + 56

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Topic2.1.2

Getting Rid of Grouping SymbolsGetting Rid of Grouping Symbols

Round UpRound Up

The distributive property is really useful — it’s always good to get rid of confusing grouping symbols whenever you can.

The main thing you need to watch out for is if you’re multiplying the contents of parentheses by a negative number — it will change the sign of everything in the parentheses.