The shaded portion of Figure shows the graph of which of the following ?

Options

A. `x(y-(2)/(3)x)gt 0`
B. `x(y-(3)/(2)x)ge 0`
C. `x(y+(3)/(2))ge 0`
D. `x(y+(3)/(2)x)le 0`

Answer

D

Explanation

First find the equation of the line to help you eliminate answer choices. The line passes through point (-2, 3) and the origin, so its equation is `y = -(3)/(2)x`.
According to the graph, you're looking for the answer choice that has the following properties : `x le 0` and `y ge - (3)/(2)x` (the shaded protion of the second quadrant)
IN ADDITION
`x ge 0` and `y le - (3)/(2)x` (the shaded portion of the fourth quadrant).
To fulfill the first set of requirements, it is necessary that `x le 0` and `y + (3)/(2)x ge 0` .
The product of these two expressions has to be negative, because the product of a negative and a positive is always negative. Therefore, `(x)(y+(3)/(2)x)le 0`. To fulfill the second set of requirements, it is necessary that `x ge 0` and `y+(3)/(2)x le 0`.
Once again, the product of these two expression has to be negative, resulting in the same inequality.
Therefore, `x(y+(3)/(2)x)le 0` fits the shaped portions of the graph.