Question

two negative integers are 8 units apart on the number line and have a product of 308. Which equation could be used to determine x, the smaller negative integer?

Answer 1

Integer are (-14) and (-22). The smallest integer is (-22).

Explanation:

Let the integer be (-x) and (-y)

(-y) - (-x) = 8 (x here is a smaller integer)

-x = b, -y = a

a - b = 8...(1)

(-a)(-b) = 308..(2)

Putting value of ab from (2) in (1):

`a=(308)/(b)`

`(308)/(b)-b=8`

`b^2-8b-308`

`b=-14,22`

`b=-x`

`-14=-x` (b=-14)

x = 14 (reject, integer asked are negative)

`b=-x`

`22=-x` (b=22)

x = -22

`a=(308)/(b)=(308)/(22)=14`

`a=14=-y`

y = -14

Integer are (-14) and (-22). The smallest integer is (-22).

Answer 2

`x = -22`

Explanation:

The formula for calculating the distance between two integers x and y is:

`| y-x | = d`

Where d is the difference between the two numbers and x is the smallest integer

In this case we know that
`d = 8` then:

`| y-x | = 8`

We also know that the product of both numbers is equal to 308.

This means that:

`xy = 308`

We know that
`x <y` and that
`x <0` and
`y <0`

then the difference of
`y-x` will always be positive, for this reason we can eliminate the absolute value of the first equation and we have that:

`y-x = 8`

and

`xy = 308`

We substitute the first equation in the second equation:

`x (x + 8) = 308`

Now we solve for x:

`x ^ 2 + 8x -308 = 0`

To factor the equation, you must look for two numbers that, by multiplying them, you get as a result -308 and by adding these numbers you get as a result 8.

These numbers are 22 and -14

Then the equation is as follows:

`(x + 22) (x-14) = 0`

The solutions are:

`x = -22`, and
`x = 14`

As we know that
`x <0` then we take the negative solution
`x = -22`

Finally we find the value of y.

`y -(-22) = 8`

`y = -22 + 8`

`y = -14`