Question

Check whether the following statements are true or false. If true, then state the property illustrated by the statement.
1. $
  \dfrac{{ - 2}}{3} + 0 = 0 + \left( {\dfrac{{ - 2}}{3}} \right) \\
    \\
 $
2. $\dfrac{{ - 4}}{5} + \dfrac{7}{8} = \dfrac{7}{8} + \left( {\dfrac{{ - 4}}{5}} \right)$

Answer 1

Hint:This can be solved by using the identity property, commutative property and associative property of addition and subtraction.

Formula used:The additive identity property is,
${\text{x}} + 0 = 0 + {\text{x}}$
The subtractive identity property is,
${\text{x}} - 0 \ne 0 - {\text{x}}$
The additive commutative property is,
${\text{x}} + {\text{y}} = {\text{y}} + {\text{x}}$
The subtractive commutative property is,
${\text{x}} - {\text{y}} \ne {\text{y}} - {\text{x}}$
Where,
${\text{x}}$and ${\text{y}}$can be any numbers

Complete step-by-step answer:
The given two equations in the question are,
$
  \dfrac{{ - 2}}{3} + 0 = 0 + \left( {\dfrac{{ - 2}}{3}} \right) \\
    \\
 $
$\dfrac{{ - 4}}{5} + \dfrac{7}{8} = \dfrac{7}{8} + \left( {\dfrac{{ - 4}}{5}} \right)$
Let take the first equation from the given question,
By using additive identity property in the LHS we get,
$ \Rightarrow \dfrac{{ - 2}}{3} + 0 = \dfrac{{ - 2}}{3}$
By using the same additive identity property in the RHS we get,
$ \Rightarrow 0 + \left( {\dfrac{{ - 2}}{3}} \right) = \dfrac{{ - 2}}{3}$
We found that the LHS and RHS of the first equation is equal.
Let take the second equation from the given question, we get
$ \Rightarrow \dfrac{{ - 4}}{5} + \dfrac{7}{8} = \dfrac{7}{8} + \left( {\dfrac{{ - 4}}{5}} \right)$
By using additive commutative property let take the LHS,
$ \Rightarrow \dfrac{{ - 4}}{5} + \dfrac{7}{8} = \dfrac{{ - 32 + 35}}{{40}}$
By solving we get,
$ \Rightarrow \dfrac{3}{{40}}$
Let take RHS of the second equation,
$ \Rightarrow \dfrac{7}{8} + \left( {\dfrac{{ - 4}}{5}} \right) = \dfrac{{35 + ( - 32)}}{{40}}$
By solving we get,
$ \Rightarrow \dfrac{3}{{40}}$
We found that the LHS and RHS of the second equation is equal.
Hence, both the equations are true by using the property of addition.
$\therefore $ The first equation is true under the additive identity property and the second equation is also true under the additive commutative property.

Note:While solving we will be getting the same value since the negative fractional number is inside the bracket. The properties will be true for addition and multiplication. For subtraction and division, the properties will be false.