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Properties of Real Numbers

Use the commutative and associative properties

Use the Commutative and Associative Properties

The order we add two numbers doesn’t affect the result. If we add $8+9$ or $9+8,$ the results are the same—they both equal 17. So, $8+9=9+8.$ The order in which we add does not matter!

Similarly, when multiplying two numbers, the order does not affect the result. If we multiply $9\cdot 8$ or $8\cdot 9$ the results are the same—they both equal 72. So, $9\cdot 8=8\cdot 9.$ The order in which we multiply does not matter!

These examples illustrate the Commutative Property.

The Commutative Property has to do with order. We subtract $9-8$ and $8-9$, and see that $9-8\ne 8-9.$ Since changing the order of the subtraction does not give the same result, we know that subtraction is not commutative.

Division is not commutative either. Since $12\div 3\ne 3\div 12,$ changing the order of the division did not give the same result. The commutative properties apply only to addition and multiplication!

Addition and multiplication are commutative.

Subtraction and division are not commutative.

\[\begin{array}{llll}(10-3)-2\ne 10-(3-2) & & & (24\div 4)\div 2\ne 24\div (4\div 2) \\ 7-2\ne 10-1 & & & \ 6\div 2\ne 24\div 2 \\ 5\ne 9 & & & \ 3\ne 12\end{array}\]

Example

Try it.

Simplify: $18p+6q+15p+5q.$

Solution

	$18p+6q+15p+5q$
Use the Commutative Property of addition to reorder so that like terms are together.	$18p+15p+6q+5q$
Add like terms.	$33p+11q$

Example

Try it.

Simplify: $(\frac{5}{13}+\frac{3}{4})+\frac{1}{4}.$

Solution

	$(\frac{5}{13}+\frac{3}{4})+\frac{1}{4}$
Notice that the last 2 terms have a common denominator, so change the grouping.	$\frac{5}{13}+(\frac{3}{4}+\frac{1}{4})$
Add in parentheses first.	$\frac{5}{13}+(\frac{4}{4})$
Simplify the fraction.	$\frac{5}{13}+1$
Add.	$1\frac{5}{13}$
Convert to an improper fraction.	$\frac{18}{13}$

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Use the Properties of Identity, Inverse, and Zero

What happens when we add 0 to any number? Adding 0 doesn’t change the value. For this reason, we call 0 the additive identity. The Identity Property of Addition that states that for any real number $a,a+0=a$ and $0+a=a.$

What happens when we multiply any number by one? Multiplying by 1 doesn’t change the value. So we call 1 the multiplicative identity. The Identity Property of Multiplication that states that for any real number $a,a\cdot 1=a$ and $1\cdot a=a.$

We summarize the Identity Properties here.

What number added to 5 gives the additive identity, 0? We know

The missing number was the opposite of the number!

We call $\text{-}a$ the additive inverse of $a.$ The opposite of a number is its additive inverse. A number and its opposite add to zero, which is the additive identity. This leads to the Inverse Property of Addition that states for any real number $a,a+(\text{-}a)=0.$

What number multiplied by $\frac{2}{3}$ gives the multiplicative identity, 1? In other words, $\frac{2}{3}$ times what results in 1? We know

Example

Try it.

Simplify: $-84n+(-73n)+84n.$

Solution

	$-84n+(-73n)+84n$
Notice that the first and third terms are opposites; use the Commutative Property of addition to re-order the terms.	$-84n+84n+(-73n)$
Add left to right.	$0+(-73n)$
Add.	$-73n$

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Simplify Expressions Using the Distributive Property

Suppose that three friends are going to the movies. They each need $9.25—that’s 9 dollars and 1 quarter—to pay for their tickets. How much money do they need all together?

You can think about the dollars separately from the quarters. They need 3 times $9 so $27 and 3 times 1 quarter, so 75 cents. In total, they need $27.75. If you think about doing the math in this way, you are using the Distributive Property.

In algebra, we use the Distributive Property to remove parentheses as we simplify expressions.

Example

Try it.

Simplify: $3(x+4).$

Solution

	$3(x+4)$
Distribute.	$3\cdot x+3\cdot 4$
Multiply.	$3x+12$

Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in would look like this:

Example

Try it.

Simplify: $8(\frac{3}{8}x+\frac{1}{4}).$

Solution


Distribute.
Multiply.

Using the Distributive Property as shown in the next example will be very useful when we solve money applications in later chapters.

Example

Try it.

Simplify: $100(0.3+0.25q).$

Solution


Distribute.
Multiply.

When we distribute a negative number, we need to be extra careful to get the signs correct!

Example

Try it.

Simplify: $-11(4-3a).$

Solution

	$-11(4-3a)$
Distribute.	$-11\cdot 4-(-11)\cdot 3a$
Multiply.	$-44-(-33a)$
Simplify.	$-44+33a$

Notice that you could also write the result as $33a-44.$ Do you know why?

In the next example, we will show how to use the Distributive Property to find the opposite of an expression.

Example

Try it.

Simplify: $\text{-}(y+5).$

Solution

	$\text{-}(y+5)$
Multiplying by $-1$ results in the opposite.	$-1(y+5)$
Distribute.	$-1\cdot y+(-1)\cdot 5$
Simplify.	$-y+(-5)$
Simplify.	$-y-5$

Example

Try it.

Simplify: $8-2(x+3)$

Solution

We follow the order of operations. Multiplication comes before subtraction, so we will distribute the 2 first and then subtract.

	$8-2(x+3)$
Distribute.	$8-2\cdot x-2\cdot 3$
Multiply.	$8-2x-6$
Combine like terms.	$-2x+2$

Condensed — the full section is in OpenStax Intermediate Algebra 2e.

Key Concepts

Commutative Property
When adding or multiplying, changing the order gives the same result

$\begin{array}{llllll}\ \text{of addition}\ \text{If}\ a,b\ \text{are real numbers, then} & & & \ a+b & = & b+a \\ \text{of multiplication}\ \text{If}\ a,b\ \text{are real numbers, then} & & & \ a\cdot b & = & b\cdot a\end{array}$

Associative Property
When adding or multiplying, changing the grouping gives the same result.

$\begin{array}{llllll}\ \text{of addition}\ \text{If}\ a,b,\ \text{and}\ c\ \text{are real numbers, then} & & & \ (a+b)+c & = & a+(b+c) \\ \text{of multiplication}\ \text{If}\ a,b,\ \text{and}\ c\ \text{are real numbers, then} & & & \ (a\cdot b)\cdot c & = & a\cdot (b\cdot c)\end{array}$

Distributive Property

$\begin{array}{llllll}\ \text{If}\ a,b,\ \text{and}\ c\ \text{are real numbers, then} & & & \ a(b+c) & = & ab+ac \\ \\ & & & \ (b+c)a & = & ba+ca \\ \\ & & & \ a(b-c) & = & ab-ac \\ \\ & & & \ (b-c)a & = & ba-ca\end{array}$

Identity Property

$\begin{array}{llll}\ \text{of addition}\ \text{For any real number}\ a\text{:} & & & \ a+0=a \\ 0\ \text{is the}\ \text{additive identity} & & & \ 0+a=a \\ \text{of multiplication}\ \text{For any real number}\ a\text{:} & & & \ a\cdot 1=a \\ 1\ \text{is the}\ \text{multiplicative identity} & & & \ 1\cdot a=a\end{array}$

Inverse Property

$\begin{array}{llll}\ \text{of addition}\ \text{For any real number}\ a, & & & \ a+(\text{-}a)=0 \\ \text{-}a\ \text{is the}\ \text{additive inverse}\ \text{of}\ a & & & \\ \text{A number and its}\ opposite\ \text{add to zero.} & & & \\ \text{of multiplication}\ \text{For any real number}\ a,a\ne 0 & & & \ a\cdot \frac{1}{a}=1 \\ \\ \frac{1}{a}\ \text{is the}\ \text{multiplicative inverse}\ \text{of}\ a & & & \\ \text{A number and its}\ reciprocal\ \text{multiply to one.} & & & \end{array}$

Properties of Zero
$\begin{array}{llll}\ \text{For any real number}\ a, & & & \ a\cdot 0=0 \\ & & & \ 0\cdot a=0 \\ \text{For any real number}\ a,a\ne 0, & & & \ \frac{0}{a}=0 \\ \text{For any real number}\ a, & & & \ \frac{a}{0}\ \text{is undefined}\end{array}$

Use the Commutative and Associative Properties

Think about adding two numbers, say 5 and 3. The order we add them doesn’t affect the result, does it?

\[\begin{array}{llll}5+3 & & & 3+5 \\ 8 & & & 8\end{array}\]\[5+3=3+5\]

The results are the same.

As we can see, the order in which we add does not matter!

What about multiplying $5\ \text{and}\ 3?$

\[\begin{array}{llll}5\cdot 3 & & & 3\cdot 5 \\ 15 & & & 15\end{array}\]\[5\cdot 3=3\cdot 5\]

Again, the results are the same!

The order in which we multiply does not matter!

These examples illustrate the commutative property. When adding or multiplying, changing the order gives the same result.

\[\begin{array}{lll}\begin{array}{ll}7-3 & \ 3-7 \\ 4 & \ -4\end{array} \\ \\ \\ 4\ne \text{-}4 \\ 7-3\ne 3-7\end{array}\]\[\begin{array}{llll} \\ \begin{array}{ll}12\div 4 & \ 4\div 12 \\ \frac{12}{4} & \ \frac{4}{12} \\ 3 & \ \frac{1}{3}\end{array} \\ 3\ne \frac{1}{3} \\ 12\div 4\ne 4\div 12\end{array}\]

Addition and multiplication are commutative.
Subtraction and Division are not commutative.

\[7+8+2\]

Add $7+8$. Add.	$\begin{array}{l}(7+8)+2 \\ 15+2 \\ 17\end{array}$
Add $8+2$. Add.	$\begin{array}{l}7+(8+2) \\ 7+10 \\ 17\end{array}$
	$(7+8)+2=7+(8+2)$

Multiply. $5\cdot \frac{1}{3}$ Multiply.	$\begin{array}{l}\ (5\cdot \frac{1}{3})\cdot 3 \\ \frac{5}{3}\cdot 3 \\ 5\end{array}$
Multiply. $\frac{1}{3}\cdot 3$. Multiply.	$\begin{array}{l}5\cdot (\frac{1}{3}\cdot 3) \\ 5\cdot 1 \\ 5\end{array}$
	$(5\cdot \frac{1}{3})\cdot 3=5\cdot (\frac{1}{3}\cdot 3)$

Example

Try it.

Simplify: $18p+6q+15p+5q.$

Solution

	$18p+6q+15p+5q$
Use the commutative property of addition to re-order so that like terms are together.	$18p+15p+6q+5q$
Add like terms.	$33p+11q$

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Use the Identity and Inverse Properties of Addition and Multiplication

What happens when we add 0 to any number? Adding 0 doesn’t change the value. For this reason, we call 0 the additive identity.

For example,

\[\begin{array}{lllllll}13+0 & & & -14+0 & & & 0+(-8) \\ 13 & & & -14 & & & -8\end{array}\]

These examples illustrate the Identity Property of Addition that states that for any real number $a,$ $a+0=a$ and $0+a=a.$

What happens when we multiply any number by one? Multiplying by 1 doesn’t change the value. So we call 1 the multiplicative identity.

For example,

\[\begin{array}{lllllll}43\cdot 1 & & & -27\cdot 1 & & & 1\cdot \frac{3}{5} \\ 43 & & & -27 & & & \frac{3}{5}\end{array}\]

These examples illustrate the Identity Property of Multiplication that states that for any real number $a,$ $a\cdot 1=a$ and $1\cdot a=a.$

We summarize the Identity Properties below.

Example

Try it.

Find the additive inverse of ⓐ $\frac{5}{8}$ ⓑ $0.6$ ⓒ $-8$ ⓓ $-\ \frac{4}{3}.$

Solution

To find the additive inverse, we find the opposite.

ⓐ The additive inverse of $\frac{5}{8}$ is the opposite of $\frac{5}{8}.$ The additive inverse of $\frac{5}{8}$ is $-\ \frac{5}{8}.$
ⓑ The additive inverse of 0.6 is the opposite of 0.6. The additive inverse of 0.6 is $-0.6.$
ⓒ The additive inverse of $-8$ is the opposite of $-8.$ We write the opposite of $-8$ as $\text{-}(-8),$ and then simplify it to 8. Therefore, the additive inverse of $-8$ is 8.
ⓓ The additive inverse of $-\ \frac{4}{3}$ is the opposite of $-\ \frac{4}{3}.$ We write this as $\text{-}(-\ \frac{4}{3}),$ and then simplify to $\frac{4}{3}.$ Thus, the additive inverse of $-\ \frac{4}{3}$ is $\frac{4}{3}.$

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Use the Properties of Zero

The identity property of addition says that when we add 0 to any number, the result is that same number. What happens when we multiply a number by 0? Multiplying by 0 makes the product equal zero.

What about division involving zero? What is $0\div 3?$ Think about a real example: If there are no cookies in the cookie jar and 3 people are to share them, how many cookies does each person get? There are no cookies to share, so each person gets 0 cookies. So,

\[0\div 3=0\]

We can check division with the related multiplication fact.

\[12\div 6=2\ \text{because}\ 2\cdot 6=12.\]

So we know $0\div 3=0$ because $0\cdot 3=0.$

Now think about dividing by zero. What is the result of dividing 4 by 0? Think about the related multiplication fact: $4\div 0=?$ means $?\cdot 0=4.$ Is there a number that multiplied by 0 gives 4? Since any real number multiplied by 0 gives 0, there is no real number that can be multiplied by 0 to obtain 4.

We conclude that there is no answer to $4\div 0$ and so we say that division by 0 is undefined.

We summarize the properties of zero below.

Example

Try it.

Simplify: ⓐ $-8\cdot 0$ ⓑ $\frac{0}{-2}$ ⓒ $\frac{-32}{0}.$

Solution

ⓐ The product of any real number and 0 is 0.	$\begin{array}{l}-8\cdot 0 \\ 0\end{array}$
ⓑ The product of any real number and 0 is 0.	$\begin{array}{l}\frac{0}{-2} \\ 0\end{array}$
ⓒ Division by 0 is undefined.	$\begin{array}{l}\frac{-32}{0} \\ \text{Undefined}\end{array}$

Example

Try it.

Simplify: ⓐ $\frac{0}{n+5},$ where $n\ne \text{-}5$ ⓑ $\frac{10-3p}{0},$ where $10-3p\ne 0.$

Solution

ⓐ Zero divided by any real number except itself is 0.	$\begin{array}{l}\frac{0}{n+5} \\ 0\end{array}$
ⓑ Division by 0 is undefined.	$\begin{array}{l}\frac{10-3p}{0} \\ \text{Undefined}\end{array}$

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Simplify Expressions Using the Distributive Property

Suppose that three friends are going to the movies. They each need $9.25—that’s 9 dollars and 1 quarter—to pay for their tickets. How much money do they need all together?

You can think about the dollars separately from the quarters. They need 3 times $9 so $27, and 3 times 1 quarter, so 75 cents. In total, they need $27.75. If you think about doing the math in this way, you are using the distributive property.

Back to our friends at the movies, we could find the total amount of money they need like this:

\[\begin{array}{lllllll} \\ \\ 3(9.25) \\ \begin{array}{lll}3(9 & + & 0.25) \\ 3(9) & + & 3(0.25) \\ 27 & + & 0.75\end{array} \\ \\ \\ 27.75\end{array}\]

In algebra, we use the distributive property to remove parentheses as we simplify expressions.

For example, if we are asked to simplify the expression $3(x+4),$ the order of operations says to work in the parentheses first. But we cannot add x and 4, since they are not like terms. So we use the distributive property, as shown in .

Example

Try it.

Simplify: $3(x+4).$

Solution

	$3(x+4)$
Distribute.	$3\cdot x+3\cdot 4$
Multiply.	$3x+12$

Some students find it helpful to draw in arrows to remind them how to use the distributive property. Then the first step in would look like this:

Example

Try it.

Simplify: $8(\frac{3}{8}x+\frac{1}{4}).$

Solution


Distribute.
Multiply.

Using the distributive property as shown in will be very useful when we solve money applications in later chapters.

Example

Try it.

Simplify: $100(0.3+0.25q).$

Solution


Distribute.
Multiply.

Example

Try it.

Simplify: $-2(4y+1).$

Solution


Distribute.
Multiply.

Example

Try it.

Simplify: $-11(4-3a).$

Solution

Distribute.
Multiply.
Simplify.

Notice that you could also write the result as $33a-44.$ Do you know why?

Example

Try it.

Simplify: $\text{-}(y+5).$

Solution

	$(y+5)$
Multiplying by −1 results in the opposite.	$-1(y+5)$
Distribute.	$-1\cdot y+(-1)\cdot 5$
Simplify.	$\text{-}y+(-5)$
	$\text{-}y-5$

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Key Concepts

Commutative Property of
- Addition: If $a,b$ are real numbers, then $a+b=b+a.$
- Multiplication: If $a,b$ are real numbers, then $a\cdot b=b\cdot a.$ When adding or multiplying, changing the order gives the same result.
Associative Property of
- Addition: If $a,b,c$ are real numbers, then $(a+b)+c=a+(b+c).$
- Multiplication: If $a,b,c$ are real numbers, then $(a\cdot b)\cdot c=a\cdot (b\cdot c).$
  When adding or multiplying, changing the grouping gives the same result.
Distributive Property: If $a,b,c$ are real numbers, then
- $a(b+c)=ab+ac$
- $(b+c)a=ba+ca$
- $a(b-c)=ab-ac$
- $(b-c)a=ba-ca$
Identity Property
- of Addition: For any real number $a\text{:}\ a+0=a\ 0+a=a$
  0 is the additive identity
- of Multiplication: For any real number $a\text{:}\ a\cdot 1=a\ 1\cdot a=a$
  $1$ is the multiplicative identity
Inverse Property
- of Addition: For any real number $a,a+(\text{-}a)=0.$ A number and its opposite add to zero. $\text{-}a$ is the additive inverse of $a.$
- of Multiplication: For any real number $a,(a\ne 0)\ a\cdot \frac{1}{a}=1.$ A number and its reciprocal multiply to one. $\frac{1}{a}$ is the multiplicative inverse of $a.$
Properties of Zero
- For any real number $a,$
  $a\cdot 0=0\ 0\cdot a=0$ – The product of any real number and 0 is 0.
- $\frac{0}{a}=0$ for $a\ne 0$ – Zero divided by any real number except zero is zero.
- $\frac{a}{0}$ is undefined – Division by zero is undefined.

Properties of Real Numbers

Use the Commutative and Associative Properties

In the following exercises, use the associative property to simplify.

Try it.

3(4x)

Solution

12x

Try it.

4(7m)

Try it.

$(y+12)+28$

Solution

$y+40$

Try it.

$(n+17)+33$

In the following exercises, simplify.

Try it.

$\frac{1}{2}+\frac{7}{8}+(-\ \frac{1}{2})$

Solution

$\frac{7}{8}$

Try it.

$\frac{2}{5}+\frac{5}{12}+(-\ \frac{2}{5})$

Try it.

$\frac{3}{20}\cdot \frac{49}{11}\cdot \frac{20}{3}$

Solution

$\frac{49}{11}$

Try it.

$\frac{13}{18}\cdot \frac{25}{7}\cdot \frac{18}{13}$

Try it.

$-24\cdot 7⋅\frac{3}{8}$

Solution

$-63$

Try it.

$-36\cdot 11\cdot \frac{4}{9}$

Try it.

$(\frac{5}{6}+\frac{8}{15})+\frac{7}{15}$

Solution

$1\frac{5}{6}$

Try it.

$(\frac{11}{12}+\frac{4}{9})+\frac{5}{9}$

Try it.

17(0.25)(4)

Solution

Try it.

36(0.2)(5)

Try it.

[2.48(12)](0.5)

Solution

14.88

Try it.

[9.731(4)](0.75)

Try it.

7(4a)

Solution

28a

Try it.

9(8w)

Try it.

$-15(5m)$

Solution

$-75m$

Try it.

$-23(2n)$

Try it.

$12(\frac{5}{6}p)$

Solution

10p

Try it.

$20(\frac{3}{5}q)$

Try it.

$43m+(-12n)+(-16m)+(-9n)$

Solution

$27m+(-21n)$

Try it.

$-22p+17q+(-35p)+(-27q)$

Try it.

$\frac{3}{8}g+\frac{1}{12}h+\frac{7}{8}g+\frac{5}{12}h$

Solution

$\frac{5}{4}g+\frac{1}{2}h$

Try it.

$\frac{5}{6}a+\frac{3}{10}b+\frac{1}{6}a+\frac{9}{10}b$

Try it.

$6.8p+9.14q+(-4.37p)+(-0.88q)$

Solution

$2.43p+8.26q$

Try it.

$9.6m+7.22n+(-2.19m)+(-0.65n)$

Use the Identity and Inverse Properties of Addition and Multiplication

In the following exercises, find the additive inverse of each number.

Try it.

ⓐ $\frac{2}{5}$ ⓑ 4.3
ⓒ $-8$
ⓓ $-\ \frac{10}{3}$

Solution

ⓐ $-\ \frac{2}{5}$ ⓑ $-4.3$ ⓒ 8 ⓓ $\frac{10}{3}$

Try it.

ⓐ $\frac{5}{9}$
ⓑ 2.1 ⓒ $-3$ ⓓ $-\ \frac{9}{5}$

Try it.

ⓐ $-\ \frac{7}{6}$ ⓑ $-0.075$ ⓒ 23 ⓓ $\frac{1}{4}$

Solution

ⓐ $\frac{7}{6}$ ⓑ 0.075 ⓒ $-23$ ⓓ $-\ \frac{1}{4}$

Try it.

ⓐ $-\ \frac{8}{3}$ ⓑ $-0.019$ ⓒ 52 ⓓ $\frac{5}{6}$

In the following exercises, find the multiplicative inverse of each number.

Try it.

ⓐ 6 ⓑ $-\ \frac{3}{4}$ ⓒ 0.7

Solution

ⓐ $\frac{1}{6}$ ⓑ $-\ \frac{4}{3}$ ⓒ $\frac{10}{7}$

Try it.

ⓐ 12 ⓑ $-\ \frac{9}{2}$ ⓒ 0.13

Try it.

ⓐ $\frac{11}{12}$ ⓑ $-1.1$ ⓒ $-4$

Solution

ⓐ $\frac{12}{11}$ ⓑ $-\ \frac{10}{11}$ ⓒ $-\ \frac{1}{4}$

Try it.

ⓐ $\frac{17}{20}$ ⓑ $-1.5$ ⓒ $-3$

Use the Properties of Zero

In the following exercises, simplify.

Try it.

$\frac{0}{6}$

Solution

Try it.

$\frac{3}{0}$

Try it.

$0\div \frac{11}{12}$

Solution

Try it.

$\frac{6}{0}$

Try it.

$\frac{0}{3}$

Solution

Try it.

$0\cdot \frac{8}{15}$

Try it.

$(-3.14)(0)$

Solution

Try it.

$\frac{\frac{1}{10}}{0}$

Mixed Practice

In the following exercises, simplify.

Try it.

$19a+44-19a$

Solution

Try it.

$27c+16-27c$

Try it.

$10(0.1d)$

Solution

Try it.

$100(0.01p)$

Try it.

$\frac{0}{u-4.99},$ where $u\ne 4.99$

Solution

Try it.

$\frac{0}{v-65.1},$ where $v\ne 65.1$

Try it.

$0\div (x-\ \frac{1}{2}),$ where $x\ne \frac{1}{2}$

Solution

Try it.

$0\div (y-\ \frac{1}{6}),$ where $y\ne \frac{1}{6}$

Try it.

$\frac{32-5a}{0},$ where
$32-5a\ne 0$

Solution

undefined

Try it.

$\frac{28-9b}{0},$ where
$28-9b\ne 0$

Try it.

$(\frac{3}{4}+\frac{9}{10}m)\div 0$ where
$\frac{3}{4}+\frac{9}{10}m\ne 0$

Solution

undefined

Try it.

$(\frac{5}{16}n-\ \frac{3}{7})\div 0$ where
$\frac{5}{16}n-\ \frac{3}{7}\ne 0$

Try it.

$15\cdot \frac{3}{5}(4d+10)$

Solution

$36d+90$

Try it.

$18\cdot \frac{5}{6}(15h+24)$

Simplify Expressions Using the Distributive Property

In the following exercises, simplify using the distributive property.

Try it.

$8(4y+9)$

Solution

$32y+72$

Try it.

$9(3w+7)$

Try it.

$6(c-13)$

Solution

$6c-78$

Try it.

$7(y-13)$

Try it.

$\frac{1}{4}(3q+12)$

Solution

$\frac{3}{4}q+3$

Try it.

$\frac{1}{5}(4m+20)$

Try it.

$9(\frac{5}{9}y-\ \frac{1}{3})$

Solution

$5y-3$

Try it.

$10(\frac{3}{10}x-\ \frac{2}{5})$

Try it.

$12(\frac{1}{4}+\frac{2}{3}r)$

Solution

$3+8r$

Try it.

$12(\frac{1}{6}+\frac{3}{4}s)$

Try it.

$r(s-18)$

Solution

$rs-18r$

Try it.

$u(v-10)$

Try it.

$(y+4)p$

Solution

$yp+4p$

Try it.

$(a+7)x$

Try it.

$-7(4p+1)$

Solution

$-28p-7$

Try it.

$-9(9a+4)$

Try it.

$-3(x-6)$

Solution

$-3x+18$

Try it.

$-4(q-7)$

Try it.

$\text{-}(3x-7)$

Solution

$-3x+7$

Try it.

$\text{-}(5p-4)$

Try it.

$16-3(y+8)$

Solution

$-3y-8$

Try it.

$18-4(x+2)$

Try it.

$4-11(3c-2)$

Solution

$-33c+26$

Try it.

$9-6(7n-5)$

Try it.

$22-(a+3)$

Solution

$\text{-}a+19$

Try it.

$8-(r-7)$

Try it.

$(5m-3)-(m+7)$

Solution

$4m-10$

Try it.

$(4y-1)-(y-2)$

Try it.

$5(2n+9)+12(n-3)$

Solution

$22n+9$

Try it.

$9(5u+8)+2(u-6)$

Try it.

$9(8x-3)-(-2)$

Solution

$72x-25$

Try it.

$4(6x-1)-(-8)$

Try it.

$14(c-1)-8(c-6)$

Solution

$6c+34$

Try it.

$11(n-7)-5(n-1)$

Try it.

$6(7y+8)-(30y-15)$

Solution

$12y+63$

Try it.

$7(3n+9)-(4n-13)$

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Simplify: $18p+6q+15p+5q.$

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$18p+6q+15p+5q$
Use the Commutative Property of addition to reorder so that like terms are together. $18p+15p+6q+5q$
Add like terms. $33p+11q$
Simplify: $23r+14s+9r+15s.$

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$32r+29s$
Simplify:$37m+21n+4m-15n.$

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$41m+6n$

Simplify: $(\frac{5}{13}+\frac{3}{4})+\frac{1}{4}.$

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	$(\frac{5}{13}+\frac{3}{4})+\frac{1}{4}$
Notice that the last 2 terms have a common denominator, so change the grouping.	$\frac{5}{13}+(\frac{3}{4}+\frac{1}{4})$
Add in parentheses first.	$\frac{5}{13}+(\frac{4}{4})$
Simplify the fraction.	$\frac{5}{13}+1$
Add.	$1\frac{5}{13}$
Convert to an improper fraction.	$\frac{18}{13}$

Simplify: $(\frac{7}{15}+\frac{5}{8})+\frac{3}{8}.$

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$1\frac{7}{15}$
Simplify: $(\frac{2}{9}+\frac{7}{12})+\frac{5}{12}.$

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$1\frac{2}{9}$
Simplify: $-84n+(-73n)+84n.$

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$-84n+(-73n)+84n$
Notice that the first and third terms are opposites; use the Commutative Property of addition to re-order the terms. $-84n+84n+(-73n)$
Add left to right. $0+(-73n)$
Add. $-73n$
Simplify: $-27a+(-48a)+27a.$

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$-48a$
Simplify: $39x+(-92x)+(-39x).$

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$-92x$

Simplify: $\frac{7}{15}\cdot \frac{8}{23}\cdot \frac{15}{7}.$

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	$\frac{7}{15}\cdot \frac{8}{23}\cdot \frac{15}{7}$
Notice the first and third terms are reciprocals, so use the Commutative Property of multiplication to re-order the factors.	$\frac{7}{15}\cdot \frac{15}{7}\cdot \frac{8}{23}$
Multiply left to right.	$1\cdot \frac{8}{23}$
Multiply.	$\frac{8}{23}$

Simplify: $\frac{9}{16}\cdot \frac{5}{49}\cdot \frac{16}{9}.$

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$\frac{5}{49}$
Simplify: $\frac{6}{17}\cdot \frac{11}{25}\cdot \frac{17}{6}.$

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$\frac{11}{25}$
Simplify: ⓐ $\frac{0}{n+5},$ where $n\ne \text{-}5$ ⓑ $\frac{10-3p}{0},$ where $10-3p\ne 0.$

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ⓐ
$\begin{array}{llll} & & & \ \frac{0}{n+5} \\ \text{Zero divided by any real number except itself is 0.} & & & \ 0\end{array}$
ⓑ
$\begin{array}{llll} & & & \ \frac{10-3p}{0} \\ \text{Division by 0 is undefined.} & & & \ \text{undefined}\end{array}$
Simplify: ⓐ $\frac{0}{m+7},$ where $m\ne \text{-}7$ ⓑ $\frac{18-6c}{0},$ where $18-6c\ne 0.$

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ⓐ 0 ⓑ undefined
Simplify: ⓐ $\frac{0}{d-4},$ where $d\ne 4$ ⓑ $\frac{15-4q}{0},$ where $15-4q\ne 0.$

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ⓐ 0 ⓑ undefined
Simplify: $3(x+4).$

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$3(x+4)$
Distribute. $3\cdot x+3\cdot 4$
Multiply. $3x+12$
Simplify: $4(x+2).$

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$4x+8$
Simplify: $6(x+7).$

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$6x+42$
Simplify: $8(\frac{3}{8}x+\frac{1}{4}).$

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Distribute.
Multiply.
Simplify: $6(\frac{5}{6}y+\frac{1}{2}).$

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$5y+3$
Simplify: $12(\frac{1}{3}n+\frac{3}{4}).$

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$4n+9$
Simplify: $100(0.3+0.25q).$

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Distribute.
Multiply.
Simplify: $100(0.7+0.15p).$

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$70+15p$
Simplify: $100(0.04+0.35d).$

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$4+35d$
Simplify: $-11(4-3a).$

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$-11(4-3a)$
Distribute. $-11\cdot 4-(-11)\cdot 3a$
Multiply. $-44-(-33a)$
Simplify. $-44+33a$
Notice that you could also write the result as $33a-44.$ Do you know why?
Simplify: $-5(2-3a).$

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$-10+15a$
Simplify: $-7(8-15y).$

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$-56+105y$
Simplify: $\text{-}(y+5).$

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$\text{-}(y+5)$
Multiplying by $-1$ results in the opposite. $-1(y+5)$
Distribute. $-1\cdot y+(-1)\cdot 5$
Simplify. $-y+(-5)$
Simplify. $-y-5$
Simplify: $\text{-}(z-11).$

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$\text{-}z+11$
Simplify: $\text{-}(x-4).$

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$\text{-}x+4$
Simplify: $8-2(x+3)$

Gosi nzaghachi

We follow the order of operations. Multiplication comes before subtraction, so we will distribute the 2 first and then subtract.
$8-2(x+3)$
Distribute. $8-2\cdot x-2\cdot 3$
Multiply. $8-2x-6$
Combine like terms. $-2x+2$
Simplify: $9-3(x+2).$

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$3-3x$
Simplify: $7x-5(x+4).$

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$2x-20$
Simplify: $4(x-8)-(x+3).$

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$4(x-8)-(x+3)$
Distribute. $4x-32-x-3$
Combine like terms. $3x-35$
Simplify: $6(x-9)-(x+12).$

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$5x-66$
Simplify: $8(x-1)-(x+5).$

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$7x-13$
$43m+(-12n)+(-16m)+(-9n)$

Gosi nzaghachi

$27m+(-21n)$
$-22p+17q+(-35p)+(-27q)$
$\frac{3}{8}g+\frac{1}{12}h+\frac{7}{8}g+\frac{5}{12}h$

Gosi nzaghachi

$\frac{5}{4}g+\frac{1}{2}h$
$\frac{5}{6}a+\frac{3}{10}b+\frac{1}{6}a+\frac{9}{10}b$

Symbols used here

\neq

not equal
The two sides are different.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Properties of Real Numbers

Use the commutative and associative properties
Use the properties of identity, inverse, and zero
Simplify expressions using the Distributive Property

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Jiri gị onwe gị

Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0), OpenStax Intermediate Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Oge Algebra

Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value

	\(18p+6q+15p+5q\)
Use the Commutative Property of addition to reorder so that like terms are together.	\(18p+15p+6q+5q\)
Add like terms.	\(33p+11q\)

	\((\frac{5}{13}+\frac{3}{4})+\frac{1}{4}\)
Notice that the last 2 terms have a common denominator, so change the grouping.	\(\frac{5}{13}+(\frac{3}{4}+\frac{1}{4})\)
Add in parentheses first.	\(\frac{5}{13}+(\frac{4}{4})\)
Simplify the fraction.	\(\frac{5}{13}+1\)
Add.	\(1\frac{5}{13}\)
Convert to an improper fraction.	\(\frac{18}{13}\)

	\(-84n+(-73n)+84n\)
Notice that the first and third terms are opposites; use the Commutative Property of addition to re-order the terms.	\(-84n+84n+(-73n)\)
Add left to right.	\(0+(-73n)\)
Add.	\(-73n\)

	\(3(x+4)\)
Distribute.	\(3\cdot x+3\cdot 4\)
Multiply.	\(3x+12\)

	\(-11(4-3a)\)
Distribute.	\(-11\cdot 4-(-11)\cdot 3a\)
Multiply.	\(-44-(-33a)\)
Simplify.	\(-44+33a\)

	\(\text{-}(y+5)\)
Multiplying by \(-1\) results in the opposite.	\(-1(y+5)\)
Distribute.	\(-1\cdot y+(-1)\cdot 5\)
Simplify.	\(-y+(-5)\)
Simplify.	\(-y-5\)

	\(8-2(x+3)\)
Distribute.	\(8-2\cdot x-2\cdot 3\)
Multiply.	\(8-2x-6\)
Combine like terms.	\(-2x+2\)

Add \(7+8\). Add.	\(\begin{array}{l}(7+8)+2 \\ 15+2 \\ 17\end{array}\)
Add \(8+2\). Add.	\(\begin{array}{l}7+(8+2) \\ 7+10 \\ 17\end{array}\)
	\((7+8)+2=7+(8+2)\)

Multiply. \(5\cdot \frac{1}{3}\) Multiply.	\(\begin{array}{l}\ (5\cdot \frac{1}{3})\cdot 3 \\ \frac{5}{3}\cdot 3 \\ 5\end{array}\)
Multiply. \(\frac{1}{3}\cdot 3\). Multiply.	\(\begin{array}{l}5\cdot (\frac{1}{3}\cdot 3) \\ 5\cdot 1 \\ 5\end{array}\)
	\((5\cdot \frac{1}{3})\cdot 3=5\cdot (\frac{1}{3}\cdot 3)\)

of addition	For any real number \(a\), \(\text{-}a\) is the additive inverse of \(a\). A number and its opposite add to zero.	\(a+(\text{-}a)=0\)
of multiplication	For any real number \(a\), \(\frac{1}{a}\) is the multiplicative inverse of \(a\). A number and its reciprocal multiply to one.	\(a\cdot \frac{1}{a}=1\)

\(\frac{0}{a}=0\)	Zero divided by any real number except itself is zero.
\(\frac{a}{0}\ \text{is undefined}\)	Division by zero is undefined.

ⓐ The product of any real number and 0 is 0.	\(\begin{array}{l}-8\cdot 0 \\ 0\end{array}\)
ⓑ The product of any real number and 0 is 0.	\(\begin{array}{l}\frac{0}{-2} \\ 0\end{array}\)
ⓒ Division by 0 is undefined.	\(\begin{array}{l}\frac{-32}{0} \\ \text{Undefined}\end{array}\)

ⓐ Zero divided by any real number except itself is 0.	\(\begin{array}{l}\frac{0}{n+5} \\ 0\end{array}\)
ⓑ Division by 0 is undefined.	\(\begin{array}{l}\frac{10-3p}{0} \\ \text{Undefined}\end{array}\)

	\((y+5)\)
Multiplying by −1 results in the opposite.	\(-1(y+5)\)
Distribute.	\(-1\cdot y+(-1)\cdot 5\)
Simplify.	\(\text{-}y+(-5)\)
	\(\text{-}y-5\)

	\(\frac{7}{15}\cdot \frac{8}{23}\cdot \frac{15}{7}\)
Notice the first and third terms are reciprocals, so use the Commutative Property of multiplication to re-order the factors.	\(\frac{7}{15}\cdot \frac{15}{7}\cdot \frac{8}{23}\)
Multiply left to right.	\(1\cdot \frac{8}{23}\)
Multiply.	\(\frac{8}{23}\)

	\(4(x-8)-(x+3)\)
Distribute.	\(4x-32-x-3\)
Combine like terms.	\(3x-35\)