Verify the following properties for the given values of a, b, c. $a=-3, b=1$ and $c=-4$.Property 1: $a\div (b+c) ≠ (a÷b) +c$
Property 2: $a\times (b+c) =(a\times b)+(a\times c)$
Property 3: $a\times (b-c)=(a\times b) -(a\times c)$

Given :

$a=-3, b=1$ and $c=-4$.

To do :

We have to verify the following properties, for the given values of a, b, c.

Property 1: $a\div (b+c)  ≠  (a÷b) +c$

Property 2: $a\times (b+c) =(a\times b)+(a\times c)$

Property 3: $a\times (b-c)=(a\times b) -(a\times c)$  

Solution :

Property 1: $a\div (b+c) ≠ (a÷b) +c$

LHS

$a÷(b+c) = -3÷(1+(-4)) = -3÷(1-4) = -3÷-3 = \frac{-3}{-3} = 1$.

RHS

$(a÷b) +c = (-3÷1)+(-4) = (-3)-4 = -7$.

$LHS ≠ RHS$

Therefore,

$a÷(b+c) ≠(a÷b) +c$.

Hence verified.

Property 2: $a\times (b+c) =(a\times b)+(a\times c)$

LHS

$a\times (b+c)  = (-3)\times (1+(-4))=(-3)\times (1-4) = (-3)\times (-3) = 9$

RHS

$(a\times b)+(a\times c) = (-3\times 1)+(-3\times -4)=(-3)+12 = 12-3 = 9$

$LHS = RHS$

Therefore,

$a\times (b+c) =(a\times b)+(a\times c)$.

Hence verified.

Property 3: $a\times (b-c)=(a\times b) -(a\times c)$ 

LHS

$a\times (b-c) = (-3) \times (1-(-4)) = (-3)\times (1+4)=(-3)\times 5 = -15$

RHS

$(a\times b) -(a\times c) = (-3\times 1)-(-3\times -4) = (-3)-(12) = -(3+12) = -15$

$LHS = RHS$

Therefore,

$a\times (b-c)=(a\times b) -(a\times c)$.

Hence verified.