You are given answer choices as follows:
A - if Quantity A will always be greater
B - if Quantity B will always be greater
C - if both quantities are equal
D - if it can't be determined
I approached this question as follows.
We are given the point q for line m and we know it passes through the origin so i decided to use point q and o to find the slope.
$0-2/0-(-3) = -2/3$
Therefore the slope equals $-2/3$ which is greater then $-1$ and therefore the answer according to me is A. However the book suggests that answer is D.
Can anyone explain where i went wrong? The book is suggesting that since line m is steeper then $-2/3$ the quantity could be less then $ - 1 $ or greater then $ - 1 $. I don't get that explanation because I thought in order to find a slope you use two points on the line as I did earlier.
Point $Q$ is not on line $m$. So when you calculate the slope of the line from $Q$ to $O$, you don't get the slope of line $m$; you get the slope of a different line. That's why answer $A$ is wrong.
The slope of the line $QO$ is $-\frac23$ as you said. The slope of line $m$, which we call $A$, must be less than this. (That is, $A = \operatorname{slope}(m) < -\frac 23$. But that is all you know. $A$ could be -1; it could be -2; it could be $-\frac45$. So you don't know whether $A$ is greater than, equal to, or less than $-1$ from the information given.