Question

Write a pair of integers whose sum gives
(i) Zero
(ii) A negative integer
(iii) An integer smaller than both integers
(iv) An integer greater than both integers
(v) An integer smaller than only one of the integers.

Answer 1

Hint: We will consider two variables say $ x $ and $ y $ . Now we will obtain an equation with these two variables according to the given condition in the problem we will assume the values of the variables are positive or negative. After assuming the values, we will check whether they satisfy the given condition or not.

Complete step by step answer:
Let us assume two variables $ x $ and $ y $ . Now the sum of the two variables is given by $ x+y $ .
(i) The sum should be zero.
If the sum of the two variables is zero, then the value $ x+y $ should be equal to zero.
 $ \begin{align}
  & \therefore x+y=0 \\
 & \Rightarrow x=-y \\
\end{align} $
From the above equation, we can say that if the sum of the two integers is zero, then the two integers are equal and having opposite signs. So, let us assume the integers as $ +5,-5 $ sum of these integers is $ +5-5=0 $ .
(ii) The sum should be a negative integer.
If the sum of the two variables is a negative integer i.e., the sum is less than zero.
 $ \begin{align}
  & \therefore x+y<0 \\
 & \Rightarrow x<-y \\
\end{align} $
From the above equation, we can say that if the sum of the two integers is a negative number, then the one integer having a greater value should have a negative sign. So, let us assume the integers $ 2,-7 $. Now the sum of the above integers is $ 2+\left( -7 \right)=2-7=-5 $ .
(iii) The sum should be smaller than both integers.
If the sum of the two variables is smaller than both the integers, then
 $ x+y < y $ and $ x+y < x $
 $ \Rightarrow x < 0 $ and $ y < 0 $ .
From the above equation, we can say that if the sum of the two integers is smaller than both the integers, then the two integers are must be less than zero. So, let us assume the integers $ -2,-7 $ . Now the sum of the above integers is $ -2+\left( -7 \right)=-2-7=-9 $ .
(iv) The sum should be greater than the two integers.
If the sum of the two variables is greater than both the integers, then
 $ x+y > y $ and $ x+y > x $
 $ \Rightarrow x > 0 $ and $ y > 0 $ .
From the above equation, we can say that if the sum of the two integers is smaller than both the integers, then the two integers are must be greater than zero. So, let us assume the integers $ 2,7 $ . Now the sum of the above integers is $ 2+7=9 $.
(v) The sum should be less than one of the integers.
If the sum of the two variables is less than one of the integers, then
 $ x+y < y $ or $ x+y < x $
 $ \Rightarrow x < 0 $ or $ y < 0 $ .
From the above equation, we can say that if the sum of the two integers is less than one of the integers, then one of the integers should be less than zero or a negative integer. So, let us assume the integers $ -2,7 $ . Now the sum of the above integers is $ -2+7=5 $.

Note:
In the second case i.e., the sum should be negative, there we have taken the condition $ x < -y $. But we can take both the negative integers for this case. So, assume the numbers $ -2,-7 $ . The sum of the integers is $ -2-7=-9 $ .