ORCCA Algebraic Properties and Simplifying Expressions

Section 1.7 Algebraic Properties and Simplifying Expressions

Objectives: PCC Course Content and Outcome Guide

MTH 60 CCOG 1.1

If we have two apples and then add three more, we have five apples. That result is the same as if we’d started with three apples and then added two more. This is a feature of numbers and arithmetic that we will formalize in this section, along with a few other features. Understanding these well will help us solve more equations later.

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Figure 1.7.1. Alternative Video Lesson

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Subsection 1.7.1 Identities and Inverses

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We start with some definitions. The number

0

is called the additive identity. It has this name because adding

0

to a number does not change that number’s “identity”. If you were playing a “game” where you needed to add something to

x,

but you didn’t want to change the value, then you would add

0 :

x + 0 = x

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Adding the additive identity to a number does not change that number.

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If the sum of two numbers is the additive identity (

0

) then those two numbers are called additive inverses of each other. Imagine playing a game where you have a number like

2,

and you need to add something to it to get

0 .

You would add

- 2 :

2 + (- 2) = 0

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It works the other way too when starting with a negative number. Now imagine starting with

- 3

and you need to add something to that to get

0 .

You would add

3 :

- 3 + 3 = 0

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The additive inverse of a number is that same number with the opposite sign. And what makes that pair of numbers special is that they add to

0 .

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We also have the special number

1,

which is the multiplicative identity. The special feature being highlighted is that when you multiply a number by

1,

it does not change that number’s “identity”. Once again, imagine you are playing a game where your job is to multiply

x

by something, but you actually do not want to change the number’s value. Then you would multiply by

1 :

x \cdot 1 = x

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If the product of two numbers is the multiplicative identity (

1

) then those two numbers are called multiplicative inverses of each other. Strategically, what would you multiply

2

by to get

1 ?

You could use the fraction

\frac{1}{2} :

2 \cdot \frac{1}{2} = 1

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Or what if you started with a more “complicated” number like

- \frac{2}{3} ?

What could you multiply that by to get

1 ?

We would use

- \frac{3}{2}

so that the negative signs cancel, and the product would be

\frac{6}{6}

which reduces to

1 :

- \frac{2}{3} \cdot (- \frac{3}{2}) = 1

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Notice what happens with the numerator and denominator swapping places. The multiplicative inverse of a number is also called its reciprocal.

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Checkpoint 1.7.2. Matching Vocabulary.

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Match each concept with the corresponding vocabulary term.

Add something but not change value
Additive identity
Add something and get $0$
Additive inverse
Multiply by something but not change value
Multiplicative identity
Multiply by something and get $1$
Multiplicative inverse

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Checkpoint 1.7.3. Matching Identities and Inverses.

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Match each description of a number on the left with a number on the right.

The additive inverse of $- 5$
$5$
The multiplicative inverse of $5$
$\frac{1}{5}$
The multiplicative inverse of $- \frac{1}{5}$ added to the additive identity
$- 5$
The multiplicative identity divided by the additive inverse of $5$
$- \frac{1}{5}$

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Subsection 1.7.2 Algebraic Properties

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Commutative Property.

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To find the area of a rectangle, multiply the length by the width. Does the result change if we multiply the width by the length?

two rectangles; the left rectangle is 3 cm wide by 2 cm high; its area is marked by Area=3*2=6; the right rectangle is 2 cm wide by 3 cm high; its area is marked by Area=2*3=6 — 🔗
Figure 1.7.4. Horizontal and Vertical Rectangles

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We can see that

3 \cdot 4 = 4 \cdot 3 .

If we write the length of a rectangle as

ℓ

and the width as

w,

we can write

ℓ w = w ℓ .

The fact that we can reverse the order of multiplication is known as the commutative property of multiplication. There is a similar property for addition, which you can see in the equation

1 + 2 = 2 + 1 .

It’s called the commutative property of addition. However, subtraction and division do not have the commutative property, because for example

2 - 1 \neq 1 - 2

and

\frac{4}{2} \neq \frac{2}{4} .

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Associative Property.

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Instead of a rectangle, consider a 3D rectangular prism with depth

d = 3 cm,

width

w = 4 cm,

and height

h = 2 cm .

To compute the volume of this prism, we multiply the depth, width, and height together. In the following figure, on the left side, we multiply the width and depth first, and then multiply the height. On the right side, we multiply the width and height first, and then multiply the depth. Either way, the final result is the same.

a rectangular prism; its length is 4 cm, width is 3 cm, and height is 2 cm; one face representing the width and depth is highlighted — Figure 1.7.5. $(3 \cdot 4) \cdot 2 = 24$

a rectangular prism; its length is 4 cm, width is 3 cm, and height is 2 cm; one face representing the width and height is highlighted — Figure 1.7.6. $3 \cdot (4 \cdot 2) = 24$

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What this shows us is

(d w) h = d (w h) .

We haven’t changed the order that the three variables are written left to right, but we are doing the actual multiplication in a different order. The order of operations requires multiplication inside the parentheses to happen first. This feature of numbers is known as the associative property of multiplication. There is a similar property for addition, which you can see in the equation

(1 + 2) + 3 = 1 + (2 + 3) .

Note again that the left-to-right order that the numbers are written is the same on either side. But the grouping symbols get us to actually do the addition in a different order. This property is called the associative property of addition. Subtraction and division do not have an associative property, because for example

(3 - 2) - 1 \neq 3 - (2 - 1)

and

(2 \div 2) \div 2 \neq 2 \div (2 \div 2) .

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Distributive Property.

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The final property we’ll explore is called the distributive property, which involves both multiplication and addition (or subtraction). To understand this property, consider what happens if we take

3

bags, and each bag contains one apple and one pear. We have the same total amount of fruit as if we’d taken a bag with

3

apples and another bag with

3

pears. Algebraically:

\begin{aligned} 3 bags, each with 1 apple and 1 pear & 3 (a + p) \\ = (bag with 3 apples) + (bag with 3 pears) & = 3 a + 3 p \end{aligned}

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It helps to think of this as “distributing” the

3

to the

a

and the

p .

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The distributive property also works with multiplication on the other side, and with subtraction:

(a + p) \cdot 3 = a \cdot 3 + p \cdot 3 3 (a - p) = 3 a - 3 p (a - p) \cdot 3 = a \cdot 3 - p \cdot 3

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although the apples and pears metaphor may be harder to make sense of.

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And the distributive property works with division too, when there is only one term in the denominator:

\frac{a + p}{3} = \frac{a}{3} + \frac{p}{3}

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Summary of Algebraic Properties.

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List 1.7.7. Algebraic Properties

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Let

a,

b,

and

c

be real numbers, variables, or algebraic expressions. Then the following properties hold:

Commutative Property of Multiplication: 🔗
$a \cdot b = b \cdot a$
Associative Property of Multiplication: 🔗
$a \cdot (b \cdot c) = (a \cdot b) \cdot c$
Commutative Property of Addition: 🔗
$a + b = b + a$
Associative Property of Addition: 🔗
$a + (b + c) = (a + b) + c$
Distributive Property: 🔗
$\begin{aligned} a (b + c) & = a b + a c & a (b - c) & = a b - a c \\ (b + c) a & = b a + c a & (b - c) a & = b a - c a \\ \frac{b + c}{a} & = \frac{b}{a} + \frac{c}{a} & \frac{b - c}{a} & = \frac{b}{a} - \frac{c}{a} \end{aligned}$

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(The last two versions require $a \neq 0 .$ )

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Practice these properties in the following exercises.

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Checkpoint 1.7.8. Matching Vocabulary.

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Match each concept with the corresponding vocabulary term.

Change the order that addition is written
Commutative Property of Addition
Change the order that multiplication is written
Commutative Property of Multiplication
Change the order that three terms are actually added together
Associative Property of Addition
Change the order that three factors are actually multiplied together
Associative Property of Multiplication
Either add two terms, then multiply; or multiply twice, then add
Distributive Property

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Checkpoint 1.7.9. Matching Properties to Examples.

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Match each algebra property with an example.

Commutative Property of Addition
$x + 4 = 4 + x$
Commutative Property of Multiplication
$x \cdot y = y \cdot x$
Associative Property of Addition
$(t + 1) + 2 = t + (1 + 2)$
Associative Property of Multiplication
$(3 y) z = 3 (y z)$
Distributive Property
$7 (m + n) = 7 m + 7 n$

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Checkpoint 1.7.10.

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(a)

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Use the commutative property of multiplication to write an equivalent expression to

53 m .

Explanation.

To use the commutative property of multiplication, change the order the two factors are multiplied:

53 m = m \cdot 53 .

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(b)

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Use the associative property of multiplication to write an equivalent expression to

3 (5 n) .

Explanation.

To use the associative property of multiplication, leave factors written in their original order, but change the grouping symbols so that a different multiplication has higher priority:

3 (5 n) = (3 \cdot 5) n .

You may further simplify by multiplying the two numbers:

\begin{aligned} 3 (5 n) & = (3 \cdot 5) n \\ = 15 n \end{aligned}

but that is going a step beyond simply using the associative property.

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(c)

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Use the commutative property of addition to write an equivalent expression to

q + 84 .

Explanation.

To use the commutative property of addition, change the order the two terms are added:

q + 84 = 84 + q .

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(d)

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Use the associative property of addition to write an equivalent expression to

x + (20 + c) .

Explanation.

To use the associative property of addition, leave terms written in their original order, but change the grouping symbols so that a different addition has higher priority:

x + (20 + c) = (x + 20) + c

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(e)

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Use the distributive property to write an equivalent expression to

3 (r + 7)

that has no grouping symbols.

Explanation.

To use the distributive property, multiply the number outside the parentheses,

3,

with each term inside the parentheses:

\begin{aligned} 3 (r + 7) & = 3 \cdot r + 3 \cdot 7 \\ = 3 r + 21 . \end{aligned}

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Subsection 1.7.3 Applying the Commutative, Associative, and Distributive Properties

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One way we use the commutative, associative, and distributive properties is to simplify algebra expressions.

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Like Terms.

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For example, how we treat like terms (see Section 2) is actually putting these algebra properties into practice. We combine like terms when we take an expression like

2 a + 3 a

and write the result as

5 a .

The drawn-out formal process actually looks like this:

\begin{aligned} 2 a + 3 a & = (2 + 3) a & distributive property \\ = 5 a & do the addition \end{aligned}

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So combining like terms is actually making use of the distributive property. In practice, it’s better for you to do this the fast way. But you can grow you understanding and appreciation for algebra if you see how the above steps break things down with one official algebra property at a time.

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Checkpoint 1.7.11.

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Where possible, simplify the following expressions by combining like terms.

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(a)

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6 c + 12 c - 5 c

Explanation.

All three terms are like terms, so they may combined. We combine them two at a time:

\begin{aligned} 6 c + 12 c - 5 c & = (6 c + 12 c) - 5 c \\ = (6 + 12) c - 5 c \\ = 18 c - 5 c \\ = (18 - 5) c = 13 c \end{aligned}

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(b)

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- 5 q^{2} - 3 q^{2}

Explanation.

The terms

- 5 q^{2}

and

- 3 q^{2}

are like terms, so we may combine them. Note we are using one of the subtraction versions of the distributive property.

\begin{aligned} - 5 q^{2} - 3 q^{2} & = (- 5 - 3) q^{2} \\ = - 8 q^{2} \end{aligned}

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(c)

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x - 5 y + 4 x

Explanation.

The terms

x

and

4 x

are like terms, while the other term is different. Using the associative and commutative properties of addition early in the process allows us to place the two like terms next to each other, and then combine them:

\begin{aligned} x - 5 y + 4 x & = x + (- 5 y) + 4 x \\ = x + ((- 5 y) + 4 x) \\ = x + (4 x + (- 5 y)) \\ = (x + 4 x) + (- 5 y) \\ = (1 + 4) x + (- 5 y) \\ = 5 x + (- 5 y) = 5 x - 5 y \end{aligned}

This walkthrough uses the algebra properties cleanly and clearly one step at a time, but if you are combining like terms more quickly, there’s nothing at all wrong with that.

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(d)

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2 x - 3 y + 4 z

Explanation.

The expression

2 x - 3 y + 4 z

cannot be simplified as there are no like terms.

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Adding Expressions.

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Consider adding an expression like

4 x + 5

to an expression like

3 x + 7 :

(4 x + 5) + (3 x + 7)

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Can we simplify this? It’s great if you already see that the parentheses in this case can be removed and it would not change the outcome:

4 x + 5 + 3 x + 7

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And then using what you know about like terms, you could conclude that this is

7 x + 12 .

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Several steps that are described above are sweeping things under the rug, taking multiple steps at once without really justifying why that is legal. Why exactly is it OK to just ignore those parentheses? Why is it ok to add

4 x

and

3 x

when they are separated with the

5

in between? With the algebra properties, we can cleanly justify why

(4 x + 5) + (3 x + 7)

simplifies to

7 x + 12,

explaining one step at a time.

\begin{aligned} (4 x + 5) + (3 x + 7) \end{aligned}

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View this as three things being added, and apply associativity of addition.

\begin{aligned} = ((4 x + 5) + 3 x) + 7 \\ = ((4 x + 5) + 3 x) + 7 \end{aligned}

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Once again apply associativity of addition. We won’t change the order anything is written, but we will move that inner pair of parentheses.

\begin{aligned} = (4 x + (5 + 3 x)) + 7 \\ = (4 x + (5 + 3 x)) + 7 \end{aligned}

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Now apply commutativity of addition.

\begin{aligned} = (4 x + (3 x + 5)) + 7 \end{aligned}

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We’ve made noteworthy progress because the

x

-terms are finally written close to each other.

\begin{aligned} = (4 x + (3 x + 5)) + 7 \end{aligned}

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Apply associativity of addition to the inner parentheses.

\begin{aligned} = ((4 x + 3 x) + 5) + 7 \\ = ((4 x + 3 x) + 5) + 7 \end{aligned}

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Apply associativity of addition again.

\begin{aligned} = (4 x + 3 x) + (5 + 7) \\ = (4 x + 3 x) + (5 + 7) \end{aligned}

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Use the distributive property.

\begin{aligned} = (4 + 3) x + (5 + 7) \end{aligned}

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And finally, we can just respect the order of operations and carry out the additions inside the parentheses.

\begin{aligned} = 7 x + 12 \end{aligned}

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That was a lot to do! So it’s worth repeating that it is good if you can more quickly simplify

(4 x + 5) + (3 x + 7)

7 x + 12 .

The above demonstrates how the algebra properties, one at a time, truly justify and validate that simplification.

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Checkpoint 1.7.12.

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Use the associative, commutative, and distributive properties to simplify the following expressions as much as possible.

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(a)

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(2 x + 3) + (4 x + 5)

Explanation.

\begin{aligned} (2 x + 3) + (4 x + 5) \\ = ((2 x + 3) + 4 x) + 5 & using the associative property of addition \\ = (2 x + (3 + 4 x)) + 5 & using the associative property of addition \\ = (2 x + (4 x + 3)) + 5 & using the commutative property of addition \\ = ((2 x + 4 x) + 3) + 5 & using the associative property of addition \\ = (2 x + 4 x) + (3 + 5) & using the associative property of addition \\ = (2 + 4) x + (3 + 5) & using the distributive property \\ = 6 x + 8 & using addition \end{aligned}

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(b)

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(- 5 x + 3) + (4 x - 7)

Explanation.

\begin{aligned} (- 5 x + 3) + (4 x - 7) \\ = ((- 5 x + 3) + 4 x) + (- 7) & using the associative property of addition \\ = (- 5 x + (3 + 4 x)) + (- 7) & using the associative property of addition \\ = (- 5 x + (4 x + 3)) + (- 7) & using the commutative property of addition \\ = ((- 5 x + 4 x) + 3) + (- 7) & using the associative property of addition \\ = (- 5 x + 4 x) + (3 + (- 7)) & using the associative property of addition \\ = (- 5 + 4) x + (3 + (- 7)) & using the distributive property \\ = - 1 x + (- 4) & using addition \\ = - x - 4 \end{aligned}

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Subsection 1.7.4 The Role of the Order of Operations

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When simplifying an expression such as

3 + 4 (5 x + 7),

we need to respect the order of operations. Since the terms inside the parentheses are not like terms, there is nothing to simplify inside the parentheses. The next highest priority operation is multiplying the

4

by the

(5 x + 7) .

This must be done before anything happens with that

3

at the far left. It is wrong to write

3 + 4 (5 x + 7) = 7 (5 x + 7),

because that would mean we treated the addition as having higher priority than the multiplication. And that just plain violates the standard order of operations.

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So the order of operations alone is not enough to simplify this expression. But the algebra properties let us rearrange things so that we can be productive. To simplify

3 + 4 (5 x + 7),

note that there is a place where we can apply the distributive property:

\begin{aligned} 3 + 4 (5 x + 7) & = 3 + (4 (5 x) + 4 (7)) \\ = 3 + (20 x + 28) \end{aligned}

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Now we can uses the commutative and associative properties.

\begin{aligned} = 3 + (28 + 20 x) \\ = (3 + 28) + 20 x \\ = 31 + 20 x \end{aligned}

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Checkpoint 1.7.13. Algebra Steps to Simplify an Expression.

Put the steps to simplify

5 + 9 (2 - 3 x)

in the correct order. Use only one algebra property or arithmetic operation in each step. It’s possible you should not use all of the steps provided here as options.

\(5+9(2-3x)\)
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\({}=14(2-3x)\) #distractor
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\({}=5+\Big(9\cdot2-9(3x)\Big)\)
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\({}=5+9(2)-3x\) #paired
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\({}=5+(18-27x)\)
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\({}=(5+18)-27x\)
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\({}=23-27x\)

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