Q48. - Algebra 2

Question

Identify the additive inverse and multiplicative inverse for each number 

$-4\frac{3}{5}$

Step-by-Step Solution

Answer

Additive inverse of $-4\frac{3}{5}$ is $4\frac{3}{5}$ and multiplicative inverse of $-4\frac{3}{5}$ is $-\frac{5}{23}$

Step 1 - Define additive inverse

Let ‘a’ be a real number such that 

$a+\left(-a\right)=0=\left(-a\right)+a$

Then ‘$-a$’ is the additive inverse of ‘a’

Step 2 - Additive inverse of - 4 3 5

Substitute $-4\frac{3}{5}$ for a 

$a=-4\frac{3}{5}$

Multiply both sides by $-1$

 $\begin{array}{c}\left(-1\right)\times a=\left(-1\right)\times \left(-4\frac{3}{5}\right)\\ -a=4\frac{3}{5}\end{array}$

Thus additive inverse of $-4\frac{3}{5}$ is $4\frac{3}{5}$

Step 3 - Define multiplicative inverse

Let ‘a’ be a real number with $a\cancel{=}0$ such that

$a·\frac{1}{a}=1=\frac{1}{a}·a$

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

Step 4 - Multiplicative inverse of - 4 3 5

Substitute $-4\frac{3}{5}$ for a 

$a=-4\frac{3}{5}$

Convert mix fraction into improper fraction

$\begin{array}{c}a=-\left(4+\frac{3}{5}\right)\\ a=-\frac{23}{5}\end{array}$

Take reciprocal on both sides

$\begin{array}{c}\frac{1}{a}=\frac{1}{-\frac{23}{5}}\\ \frac{1}{a}=-\frac{5}{23}\end{array}$

Thus multiplicative inverse of $-4\frac{3}{5}$ is $-\frac{5}{23}$