Algebraic Expressions, Equations, and Inequalities

Properties of Real Numbers

Properties of Addition

The associative, commutative, and identity properties of addition can be used to simplify expressions with real numbers. The additive identity is zero, and the additive inverse, or opposite, of a real number aa is a-a.

Three important properties of addition can be used to simplify expressions.

The commutative property of addition states that when adding real numbers, the sum does not change based on the order of the numbers:
a+b=b+aa+b=b+a
The associative property of addition states that when adding three or more numbers, the sum does not change based on the way the numbers are grouped:
a+(b+c)=(a+b)+ca+(b+c)=(a+b)+c
The identity property of addition states that when adding zero to any any given number, the sum is the given number itself:
a+0=aa+0=a
The additive identity is the number zero, which has the property that a+0=aa+0=a for any real number aa. This is stated as the identity property of addition, which is the sum of zero and any number is the given number:
a+0=aa+0=a
For a real number aa, the additive inverse is the number a-a. It is also called the opposite of aa. The sum of a number and its additive inverse is the additive identity, zero. This value is called the inverse, or opposite, because it reverses the addition. For any number, when adding aa and then adding a-a, the result is the original number:
b+a+(a)=bb+a+(-a)=b

Addition Properties

Commutative Property of Addition Associative Property of Addition Identity Property of Addition
Order does not affect the sum. Grouping does not affect the sum. Adding zero does not affect the sum.
a+b=b+aa+b=b+a
a+(b+c)=(a+b)+ca+(b+c)=(a+b)+c
a+0=aa+0=a
9+5=5+99+5=5+9
5+(7+6)=(5+7)+65+(7+6)=(5+7)+6
26+0=2626+0=26

Additive Inverses

Value Additive Inverse Number Plus Inverse
aa
a-a
a+(a)=0a+(-a)=0
9797
97-97
97+(97)=097+(-97)=0
45-\frac{4}{5}
(45)=45-\left(-\frac{4}{5}\right)=\frac{4}{5}
45+45=0-\frac{4}{5}+\frac{4}{5}=0
00
0=0-0=0
0+0=00+0=0

Properties of Multiplication

The associative, commutative, and identity properties of multiplication can be used to simplify expressions with real numbers. The multiplicative identity is 1, and the multiplicative inverse, or reciprocal, of a real number aa is 1a\frac{1}{a}.

The properties of multiplication are similar to the properties of addition and are also used to simplify expressions.

The commutative property of multiplication states that when multiplying real numbers, the product does not change based on the order of the numbers:
ab=baab=ba
The associative property of multiplication states that when multiplying three or more numbers, the product does not change based on the way the numbers are grouped:
a(bc)=(ab)ca(bc)=(ab)c
The multiplicative identity is the number 1, which has the property that a1=aa\cdot1=a for any real number aa. This is stated as the identity property of multiplication, which is the product of 1 and any number is the given number:
a1=aa\cdot1=a
For a nonzero real number aa, the multiplicative inverse is the number 1a\frac{1}{a}. It is also called the reciprocal of aa. The product of a number and its multiplicative inverse is the multiplicative identity, 1. This value is called the inverse because it reverses the multiplication. For any number, when multiplying by aa and then multiplying by 1a\frac{1}{a}, the result is the original number:
b(a)(1a)=bb(a)\left(\frac{1}{a}\right)=b
The multiplicative inverse can be applied to any value except zero, which produces an undefined multiplicative inverse. In addition, multiplying any value by zero does not result in 1, which means that the multiplicative identity does not apply to zero.

Multiplication Properties

Commutative Property of Multiplication Associative Property of Multiplication Identity Property of Multiplication
Order does not affect the product. Grouping does not affect the product. Multiplying by 1 does not affect the product.
ab=baab=ba
a(bc)=(ab)ca(bc)=(ab)c
a1=aa\cdot1=a
412=1244\cdot12=12\cdot4
3(25)=(32)53(2\cdot5)=(3\cdot2)5
71=77\cdot1=7

Multiplicative Inverses

Value Multiplicative Inverse Number Times Inverse
aa
1a\frac{1}{a}
a1aa\cdot\frac{1}{a}
88
18\frac{1}{8}
818=18\cdot\frac{1}{8}=1
3-3
13-\frac{1}{3}
3(13)=1-3\cdot\left(-\frac{1}{3}\right)=1
23\frac{2}{3}
1(23)=32\frac{1}{\left(\frac{2}{3}\right)}=\frac{3}{2}
2332=1\frac{2}{3}\cdot\frac{3}{2}=1
00
10\frac{1}{0} is undefined. So, there is no multiplicative inverse of zero. There is no number that can be multiplied by zero to produce 1.

Distributive Property

The distributive property of multiplication over addition can be used to simplify expressions with real numbers.

The distributive property states that multiplying an expression by a sum is the same as multiplying the expression by each term in the sum and then adding the products.

The value that is multiplied can be on the left side of the expression, such as:
a(b+c)a(b+c)
It can also be on the right side, such as:
(a+b)c(a+b)c
Multiplying an expression by one or more terms in a sum is called distributing. For example:
a(b+c)a(b+c)
In the expression, the aa can be distributed over the sum, resulting in:
ac+bcac+bc
For instance, simplify the expression:
2(4+6)2(4+6)
The addition can be performed first, or the 2 can be distributed over the sum. The result is the same.

Distributive Property of Multiplication over Addition

Add, then multiply. Multiply, then add.
2(4+6)2(10)20\begin{gathered}2(4+6)\\2(10)\\20\end{gathered}
2(4+6)2(4)+2(6)8+1220\begin{gathered}2(4+6)\\2(4)+2(6)\\8+12\\20\end{gathered}

Step-By-Step Example
Applying the Distributive Property
Simplify the expression:
3x(4x+5y7)3x(4x+5y-7)
Step 1
Apply the distributive property. Multiply 3x3x by each term in the sum.
3x(4x+5y7)3x(4x)+3x(5y)3x(7)\begin{gathered}3x(4x+5y-7)\\3x(4x)+3x(5y)-3x(7)\end{gathered}
Step 2
Simplify each product.
3x(4x)+3x(5y)3x(7)12x2+15xy21x\begin{gathered}3x(4x)+3x(5y)-3x(7)\\12x^2+15xy-21x\end{gathered}
Solution
The expression simplifies to:
12x2+15xy21x12x^2+15xy-21x

To simplify an expression of the form (a+b)(c+d)(a+b)(c+d), the distributive property is applied twice. First, distribute c+dc+d over the sum a+ba+b:
(a+b)(c+d)a(c+d)+b(c+d)\begin{gathered}(a+b)(c+d) \\ a(c+d)+b(c+d)\end{gathered}
Next, distribute aa and bb over the sum c+dc+d:
a(c+d)+b(c+d)ac+ad+bc+bd\begin{gathered} a(c+d)+b(c+d)\\ac+ad+bc+bd\end{gathered}
The memory aid FOIL can be used to remember the order of multiplying terms in expressions using the form:
(a+b)(c+d)(a+b)(c+d)
  • The term acac is the product of the First terms of (a+b)(c+d)({\color{#c42126}a}+b)({\color{#c42126}c}+d).
  • The term adad is the product of the Outside terms of (a+b)(c+d)({\color{#c42126}a}+b)(c+{\color{#c42126}d}).
  • The term bcbc is the product of the Inside terms of (a+b)(c+d)(a+{\color{#c42126}b})({\color{#c42126}c}+d).

The term bdbd is the product of the Last terms of (a+b)(c+d)(a+{\color{#c42126}b})(c+{\color{#c42126}d}).

Step-By-Step Example
Distributing an Expression to Another Expression
Simplify the expression:
(x+4)(x+1)(x+4)(x+1)
Step 1
Distribute x+1x+1 over the sum x+4x+4.
(x+4)(x+1)x(x+1)+4(x+1)\begin{gathered}({\color{#c42126} x+4})(x+1)\\{\color{#c42126} {x}}(x+1){\color{#c42126} {\;+\;4}}(x+1)\end{gathered}
Step 2
Distribute xx and 44 over the sum x+1x+1. The result is the products of the first, outside, inside, and last terms of the expressions
x(x+1)+4(x+1)x2+x+4x+4\begin{gathered}x(x+1)+4(x+1)\\x^2+x+4x+4\end{gathered}
Solution
Simplify by combining like terms xx and 4x4x.
x2+x+4x+1x2+5x+4\begin{gathered}x^2+x+4x+1\\x^2+5x+4\end{gathered}

The distributive property can also be used to rewrite expressions by factoring out a common factor.
6y24(6)(y)(6)(4)6(y4)\begin{gathered}6y-24\\({\color{#c42126} 6})(y)-({\color{#c42126} 6})(4)\\{\color{#c42126}{6}}(y-4)\end{gathered}
The common factor in the expression is 6. Note that the original expression, which is also the simplified form, and the factored form of the expression are equivalent:
6y246(y4)\begin{gathered}{6y-24}\\{6(y-4)}\end{gathered}
Factoring is not the same as simplifying an expression, but it may be used as a step in simplifying or in finding solutions of an equation.

Applying Properties of Operations

Expressions can be simplified by combining the properties of operations with the order of operations.

If an expression has more than one operation, performing the operations in a different order can produce different results. So, there is an agreed-upon order to ensure the result is always the same. The order of operations is a set of rules indicating which calculations to perform first to simplify a mathematical expression.

1. Simplify expressions within grouping symbols, such as parentheses ()( ), brackets [][ ], or braces {}\{ \}, or expressions in the numerator or denominator of a fraction or under a radical. For nested grouping symbols {[()]}\{ [ ( ) ] \}, start with the innermost set, and work to the outside.

2. Simplify any exponents or radicals in the expression.

3. Multiply and divide, working from left to right across the expression.

4. Add and subtract, working from left to right across the expression.

Properties of operations can be used with the order of operations to simplify expressions.

Step-By-Step Example
Simplifying Expressions with Properties of Operations
Simplify the expression:
12+5(211)+92\frac{1}{2}+5(2\cdot11)+\frac{\sqrt{9}}{2}
Step 1
The first step in the order of operations is to evaluate within grouping symbols. However, the associative property of multiplication can be used to rewrite the expression in a way that is easier to simplify.
12+5(211)+9212+(52)11+92\begin{gathered}\frac{1}{2}+{\color{#c42126}{5(2\cdot11)}}+\frac{\sqrt{9}}{2}\\\frac{1}{2}+{\color{#c42126}{(5\cdot2)11}}+\frac{\sqrt{9}}{2}\end{gathered}
Step 2
Now, evaluate the expression within the grouping symbols.
12+(52)11+9212+1011+92\begin{gathered}\frac{1}{2}+({\color{#c42126}{5\cdot2}})11+\frac{\sqrt{9}}{2}\\\frac{1}{2}+{\color{#c42126}{10}}\cdot11+\frac{\sqrt{9}}{2}\end{gathered}
Step 3
Simplify exponents and radicals.
12+1011+9212+1011+32\begin{gathered}\frac{1}{2}+10\cdot11+\frac{{\color{#c42126}{\sqrt{9}}}}{2}\\\frac{1}{2}+10\cdot11+\frac{{\color{#c42126}{3}}}{2}\end{gathered}
Step 4
The next step is to multiply.
12+1011+3212+110+32\begin{gathered}\frac{1}{2}+{\color{#c42126}{10\cdot11}}+\frac{3}{2}\\\frac{1}{2}+{\color{#c42126}{110}}+\frac{3}{2}\end{gathered}
Step 5
Use the commutative property of addition to rewrite the expression.
12+110+3212+32+110\begin{gathered}\frac{1}{2}+{\color{#c42126}{110+\frac{3}{2}}}\\\frac{1}{2}+{\color{#c42126}{\frac{3}{2}+110}}\end{gathered}
Solution
Add from left to right.
12+32+1102+110112\begin{gathered}{\color{#c42126}{\frac{1}{2}+\frac{3}{2}}}+110\\{\color{#c42126}{2}}+110\\112\end{gathered}