Question

(a)Write a pair of integers and a positive integer whose sum is $-5$\[\]
(b)Write a negative integer and a positive integer whose sum is 8\[\]
(c) )Write a negative integer and a positive integer whose difference is $-3$.\[\]

Answer 1

Hint: We recall the definitions of positive integers, negative integers and how two they are added with the use of absolute value. We use the fact that the sum of two integers is positive then the integer with larger absolute value is positive and if the sum is negative then the integer with larger absolute value is negative to solve part (i) and (ii). We solve part (iii) by taking two numbers close 0 whose difference is $-3$. \[\]

Complete step-by-step answer:
We know that the set of integers denoted by letter $Z$ and in list from written with negative integers(integers less than 0), 0 and positive integers(integers greater than 0) as
\[Z=\left\{ ...-3,,-2,-1,0,1,2,3... \right\}\]
We know that the absolute value of an integer is its positive difference with 0. Difference is the result of subtraction from greater integer to smaller integer. The absolute value of an integer $x$ is
\[\left| x \right|=\left\{ \begin{matrix}
   x-0=x & \text{if }x\ge 0 \\
   0-x=-x & \text{if }x < 0 \\
\end{matrix} \right.\]
The arithmetic operation of addition between two positive integers is the same as the natural number. We add one positive integer and one negative integer by subtracting smaller absolute value from larger absolute value. If $a,b$ are positive integers then
\[a+\left( -b \right)=\left\{ \begin{matrix}
   \left| a \right|-\left| b \right| & \text{if }\left| a \right|\ge \left| b \right| \\
   \left| b \right|-\left| a \right| & \text{if }\left| a \right| < \left| b \right| \\
\end{matrix} \right.\]
If the sum of two integers is positive then the integer with larger absolute value is positive and if the sum is negative then the integer with larger absolute value is negative.
The substation between any two integers $x,y$ is defined as
\[x-y=x+\left( -y \right)\]

 (i) We have to write a pair of integers and a positive integer whose sum is $-5$. As the sum is a negative integer the one of the numbers must be negative and also with larger absolute value than absolute value of the sum of the other two. Let the negative integer be $-12$ and to have a sum of $-5$ with three integers; we have to add 7. So the sum of pairs of integers is 7. We can chose the a pair of integers as $\left( 3,4 \right)$ and have the answer as \[\left( 3,4 \right),-12\]
(ii) We have to write a negative integer and a positive integer whose sum is 8. We see that as the sum 8 is positive the positive integer will have larger absolute value. Let that the positive integer be 12 and then we need to add $-4$ to get the sum 8. So the answer is
\[12,-4\]
(iii) We have written a negative integer and a positive integer whose difference is $-3$. S both the integers will be around 0 as absolute value of $-3$ is $-\left( -3 \right)=3$.Let the negative integer be $-1$ and we subtract 2 to get the difference$-3$ which means $-1-2=-3$ and hence the answer is
\[-1,2\]

Note: The integers are solutions of equation $x+a=0$ where $a$ is the whole number. We find the sum of two negative numbers by adding their absolute values and the adding negative sign on the sum that is$\left( -a \right)+\left( -b \right)=-\left( \left| a \right|+\left| b \right| \right)$. The absolute value also gives us the distance of the integer from 0 in the number line. There will be an infinite number of solutions for part(i) and (ii) only another pair of solutions which we can find for part (iii) is $\left( -2,1 \right)$.