1. Getting Rid of Grouping Symbols
Topic 2.1.2

2. Getting Rid of Grouping Symbols
Topic
2.1.2
Getting 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

3. Getting Rid of Grouping Symbols
Topic
2.1.2
Getting Rid of Grouping Symbols
You already saw the distributive property in Topic
For example: 3(b + c) = 3b + 3c
In this Topic you’ll simplify expressions by using the distributive property to get rid of grouping symbols.

4. Getting Rid of Grouping Symbols
Topic
2.1.2
Getting Rid of Grouping Symbols
The Distributive Property Removes 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.

5. Getting Rid of Grouping Symbols
Topic
2.1.2
Getting Rid of Grouping Symbols
Example 1
Simplify 5(3x + 2) + 2(2x – 1).
Solution
5(3x + 2) + 2(2x – 1)
Given expression
= 15x x – 2
Distributive property
= 15x + 4x + 10 – 2
Commutative property of addition
= 19x + 8
Solution follows…

6. Getting Rid of Grouping Symbols
Topic
2.1.2
Getting Rid of Grouping Symbols
Guided Practice
In Exercises 1–7, simplify the following expressions: 1.
2(4x + 5) + 8
8x + 18 2.
12(5a – 8) + 4x + 3
60a + 4x – 93 3.
6(2j + 3c) + 8(5c + 4z)
12j + 58c + 32z 4.
10(x + 2) + 7(3 – 4x)
–18x + 41 5.
6(a – b) + 4(2b – 3)
6a + 2b – 12 6.
5(3x + 4) + 3(4x + 10) + 2(8x + 9)
43x + 68 7.
8(2n – 3) + 9(4n – 5) + 4(3n + 7)
64n – 41
Solution follows…

7. Getting Rid of Grouping Symbols
Topic
2.1.2
Getting Rid of Grouping Symbols
Take Care when Multiplying by a Negative Number
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.

8. Getting Rid of Grouping Symbols
Topic
2.1.2
Getting Rid of Grouping Symbols
Example 2
Simplify the following:
a) –7(2x + 1) b) –6(–x – 3) c) –3(5x – 4)
Solutions a)
The +2x and +1 become negative.
–7(2x + 1) = –14x – 7 b)
The two negative terms inside the grouping symbols are multiplied by the negative term outside. They both become positive.
–6(–x – 3) = 6x + 18 c)
–3(5x – 4) = –15x + 12
Solution follows…

9. Getting Rid of Grouping Symbols
Topic
2.1.2
Getting Rid of Grouping Symbols
Example 3
Simplify the expression 4(2x – 1) – 5(x – 2). Show your steps.
Solution
4(2x – 1) – 5(x – 2)
Given expression
= 8x – 4 – 5x + 10
Distributive property
= 8x – 5x –
Commutative property of addition
= 3x + 6
Solution follows…

10. Getting Rid of Grouping Symbols
Topic
2.1.2
Getting Rid of Grouping Symbols
Guided Practice
In Exercises 8–13, simplify each algebraic expression:
8. –2(5a – 3c)
–10a + 6c
9. –8(3c – 2)
–24c + 16
10. –2(–3x – 4) + 4(6 – 2x)
–2x + 32
11. 7(2a + 9) – 4(a + 11)
10a + 19
12. –8(2y + 4) – 5(y + 4)
–21y – 52
13. –2 n + 2 – n – 4 1 2 3
–2n + 8
Solution follows…

11. Getting Rid of Grouping Symbols
Topic
2.1.2
Getting Rid of Grouping Symbols
Guided Practice
14. Simplify 12(2n – 7) – 9(3 – 4n) + 6(4x – 9).
60n + 24x – 165
15. Simplify 5(x – 2) – 7(–4x + 3) – 3(–2x).
39x – 31
Solution follows…

12. Getting Rid of Grouping Symbols
Topic
2.1.2
Getting Rid of Grouping Symbols
Independent Practice
In Exercises 1–7, simplify the algebraic expressions:
1. –4(a + 2b)
–4a – 8b
2. 3(2n + 4) + n(–4)
2n + 12
3. 5(3b – 2q) – (3q + 4)
15b – 13q – 4
4. –9(2 + 3b) + 3(3 – 2b)
–33b – 9
5. –17(3a – 5b) – 4(x + 3)
–51a + 85b – 4x – 12
6. –2(10n – 4x) + 2(n + 6x) – 3(7x – 2n)
–12n – x
7. –7(3p – 9q) – 4(25q – 3r) + 12(5p + 8)
39p – 37q + 12r + 96
Solution follows…

13. Getting Rid of Grouping Symbols
Topic
2.1.2
Getting Rid of Grouping Symbols
Independent Practice
In Exercises 8–12, find and simplify an expression for the perimeter of the shape shown. 9. 8.
10.
P = 6x + 3
P = 8x – 5
P = 8 – 2x
11.
12.
P = 14n – 6
P = 60 – 18n
Solution follows…

14. Getting Rid of Grouping Symbols
Topic
2.1.2
Getting Rid of Grouping Symbols
Independent Practice
In Exercises 13–15, simplify the algebraic expressions:
(3x + 4) – (4x – 8) 3 8 1 2 7 8
– x + 11 2
14. –0.1(25n + 46) – 0.8(16n – 5) + 0.4(3 – 4n)
–16.9n + 0.6
15. – (5 – 9x) – (2x – 4) – 0.2(–7x – 10) 1 3 8 5 6
4.15x +
Solution follows…

15. Getting Rid of Grouping Symbols
Topic
2.1.2
Getting Rid of Grouping Symbols
Independent Practice
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.
14x + 19
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.
18x + 56
Solution follows…

16. Getting Rid of Grouping Symbols
Topic
2.1.2
Getting Rid of Grouping Symbols
Round 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.