Step-by-step explanation:
We are to get the expression for the following statements;
1) The difference of nine times a number x and the quotient of that number and 5.
The product of nine and a number x is expressed as;
![=9 \times x\\= 9x](https://tex.z-dn.net/?f=%3D9%20%5Ctimes%20x%5C%5C%3D%209x)
The quotient of that number and 5.
![= \frac{x}{5}](https://tex.z-dn.net/?f=%3D%20%5Cfrac%7Bx%7D%7B5%7D)
The difference between both expression;
![9x - \frac{x}{5}](https://tex.z-dn.net/?f=9x%20-%20%5Cfrac%7Bx%7D%7B5%7D)
Hence, the difference of nine times a number x and the quotient of that number and 5 is expressed as ![9x - \frac{x}{5}](https://tex.z-dn.net/?f=9x%20-%20%5Cfrac%7Bx%7D%7B5%7D)
2) Eight more than the quotient of twelve and a number n
Quotient of twelve and a number n is expressed as:
![\frac{n}{12}](https://tex.z-dn.net/?f=%5Cfrac%7Bn%7D%7B12%7D)
Eight more than the resulting function is;
![\frac{n}{12}+8](https://tex.z-dn.net/?f=%5Cfrac%7Bn%7D%7B12%7D%2B8)
Hence eight more than the quotient of twelve and a number n is expressed as ![\frac{n}{12}+8](https://tex.z-dn.net/?f=%5Cfrac%7Bn%7D%7B12%7D%2B8)
3) The product of a number and the quantity 'six minus the number' plus the quotient of eight and the number.
Let the number be x:
six minus the number is expressed as;
![6-x](https://tex.z-dn.net/?f=6-x)
product of a number x and the quantity 'six minus the number is;
![x(6-x)](https://tex.z-dn.net/?f=x%286-x%29)
quotient of eight and the number is;
![\frac{8}{x}](https://tex.z-dn.net/?f=%5Cfrac%7B8%7D%7Bx%7D)
Taking the resulting sum of the last two expression
![x(6-x) + 8x](https://tex.z-dn.net/?f=x%286-x%29%20%2B%208x)
Hence the product of a number and the quantity 'six minus the number' plus the quotient of eight and the number is expressed as;
![x(6-x) + 8x](https://tex.z-dn.net/?f=x%286-x%29%20%2B%208x)
4) Sum of three consecutive even integers. 2x + (2x + 2) + (2x + 4).
Let the first even number be 2x
The consecutive even numbers are gotten by adding 2 to the preceding number. The two consecutive even integers are 2x+2 and 2x+2+2
the sum of three consecutive even integers is expressed as;
![= 2x +(2x+2)+(2x+2+2)\\= 2x+(2x+2)+(2x+4)](https://tex.z-dn.net/?f=%3D%202x%20%2B%282x%2B2%29%2B%282x%2B2%2B2%29%5C%5C%3D%20%202x%2B%282x%2B2%29%2B%282x%2B4%29)