Question

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

• A. $x(y-2x)\ge 0$
• B. $x(y-2x)\le 0$
• C. $x(y+\frac{1}{2}x)\ge 0$
• D. $x(y-\frac{1}{2}x)\le 0$

Answer 1

The correct option is A $x(y-2x)\ge 0$

Each of the answer choices in the form of the product of two factors on the left and a $"\ge 0"\le 0"$ on the right.
The product will be negative when the two factors have opposite signs, and it will be positive when the factors have the same sign. Choice (1), for example, has $a"\ge 0"$, so you'll be looking other factors to have the same sign.
Either: $x\ge 0$ and $y-2x\ge 0\Rightarrow x\ge 0$ and $y\ge 2x$ or $x\le 0$ and $y-2x\le 0\Rightarrow x\le 0$ and $y\le 2x$. The graph of $x\ge 0$ and $y\ge 2x$ looks like this :

The graph of $x\le 0$ and $y\le 2x$ looks like this

Together they make the graph in the figure.