Question

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

• A. \( x (y - 2x) \geq 0 \)
• B. \( x (y - 2x) \leq 0 \)
• C. \( x \left( y + \frac{x}{2} \right) \geq 0 \)
• D. \( x \left( y - \frac{x}{2} \right) \leq 0 \)

Hint:

When identifying inequalities from shaded graphs, check above/below line regions separately for \(x > 0\) and \(x < 0\).

Answer 1

The Correct Option is A

From the figure: - The line shown is \(y = 2x\) (slope \(2\), passing through the origin). - The shaded area is such that:
- For \(x > 0\), the region is above the line \(y = 2x\).
- For \(x > 0\), the region is below the line \(y = 2x\).
This pattern matches the inequality: \[ x (y - 2x) \geq 0 \] Check: 1. If \(x > 0\): \(y - 2x \geq 0 \Rightarrow y \geq 2x\) → matches shaded region on right.
2. If \(x < 0\): \(y - 2x \leq 0 \Rightarrow y \leq 2x\) → matches shaded region on left.
Thus, the correct inequality is: \[ {x (y - 2x) \geq 0} \]