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: [latex]3x+15 - 3x[/latex]

Solution:

[latex]3x+15 - 3x[/latex]
Notice the additive inverses, [latex]3x[/latex] and [latex]-3x[/latex] . [latex]0+15[/latex]
Add. [latex]15[/latex]

try it

example

Simplify: [latex]4\left(0.25q\right)[/latex]

try it

example

Simplify: [latex]{\Large\frac{0}{n+5}}[/latex] , where [latex]n\ne -5[/latex]

try it

example

Simplify: [latex]{\Large\frac{10 - 3p}{0}}[/latex].

try it

example

Simplify: [latex]{\Large\frac{3}{4}}\cdot {\Large\frac{4}{3}}\left(6x+12\right)[/latex].

try it

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

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

[latex]\frac{1}{a}[/latex] is the multiplicative inverse of [latex]a[/latex]
For any real number a, [latex]a+\text{(}\text{-}\mathit{\text{a}}\text{)}=0[/latex] [latex]a\cdot 1a=1[/latex]
Distributive Property

If [latex]a,b,c[/latex] are real numbers, then [latex]a\left(b+c\right)=ab+ac[/latex]
Properties of Zero
For any real number a, [latex]\begin{array}{l}a\cdot 0=0\\ 0\cdot a=0\end{array}[/latex]
For any real number [latex]a,a\ne 0[/latex] [latex]{\Large\frac{0}{a}}=0[/latex]

[latex]{\Large\frac{a}{0}}[/latex] is undefined