## Simplifying Expressions Using the Properties of Identities, Inverses, and Zero

### Learning Outcomes

• Simplify algebraic expressions using identity, inverse and zero properties
• Identify which property(ies) to use to simplify an algebraic expression

## Simplify Expressions using the Properties of Identities, Inverses, and Zero

We will now practice using the properties of identities, inverses, and zero to simplify expressions.

### example

Simplify: $3x+15 - 3x$

Solution:

 $3x+15 - 3x$ Notice the additive inverses, $3x$ and $-3x$ . $0+15$ Add. $15$

### example

Simplify: $4\left(0.25q\right)$

### example

Simplify: ${\Large\frac{0}{n+5}}$ , where $n\ne -5$

### example

Simplify: ${\Large\frac{10 - 3p}{0}}$.

### example

Simplify: ${\Large\frac{3}{4}}\cdot {\Large\frac{4}{3}}\left(6x+12\right)$.

### try it

All the properties of real numbers we have used in this chapter are summarized in the table below.

Properties of Real Numbers
Commutative Property
If a and b are real numbers then… $a+b=b+a$ $a\cdot b=b\cdot a$
Associative Property
If a, b, and c are real numbers then… $\left(a+b\right)+c=a+\left(b+c\right)$ $\left(a\cdot b\right)\cdot c=a\cdot \left(b\cdot c\right)$
Identity Property $0$ is the additive identity $1$ is the multiplicative identity
For any real number a, $\begin{array}{l}a+0=a\\ 0+a=a\end{array}$ $\begin{array}{l}a\cdot 1=a\\ 1\cdot a=a\end{array}$
Inverse Property $-\mathit{\text{a}}$ is the additive inverse of $a$ $a,a\ne 0$

$\frac{1}{a}$ is the multiplicative inverse of $a$

For any real number a, $a+\text{(}\text{-}\mathit{\text{a}}\text{)}=0$ $a\cdot 1a=1$
Distributive Property

If $a,b,c$ are real numbers, then $a\left(b+c\right)=ab+ac$

Properties of Zero
For any real number a, $\begin{array}{l}a\cdot 0=0\\ 0\cdot a=0\end{array}$
For any real number $a,a\ne 0$ ${\Large\frac{0}{a}}=0$

${\Large\frac{a}{0}}$ is undefined