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?

Accepted Solution

Answer:[tex]x = -22[/tex]Step-by-step explanation:The formula for calculating the distance between two integers x and y is:[tex]| y-x | = d[/tex]Where d is the difference between the two numbers and x is the smallest integerIn this case we know that [tex]d = 8[/tex] then:[tex]| y-x | = 8[/tex]We also know that the product of both numbers is equal to 308.This means that:[tex]xy = 308[/tex]We know that [tex]x <y[/tex] and that [tex]x <0[/tex] and [tex]y <0[/tex]then the difference of [tex]y-x[/tex] will always be positive, for this reason we can eliminate the absolute value of the first equation and we have that:[tex]y-x = 8[/tex]and[tex]xy = 308[/tex]We substitute the first equation in the second equation:[tex]x (x + 8) = 308[/tex]Now we solve for x:[tex]x ^ 2 + 8x -308 = 0[/tex]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 -14Then the equation is as follows:[tex](x + 22) (x-14) = 0[/tex]The solutions are:[tex]x = -22[/tex], and [tex]x = 14[/tex]As we know that [tex]x <0[/tex] then we take the negative solution [tex]x = -22[/tex]Finally we find the value of y.[tex]y -(-22) = 8[/tex][tex]y = -22 + 8[/tex][tex]y = -14[/tex]