Problem 78 - Precalculus Essentials

Question

List the quadrant or quadrants satisfying each condition. $$x^{3}<0 \text { and } y^{3}>0$$

Step-by-Step Solution

Answer

The conditions \(x^3<0\) and \(y^3>0\) are satisfied in the second quadrant

Step 1: Understanding the conditions

Let's first analyze each conditions separately. The condition \(x^3<0\) implies that x is negative since the cube of any negative number is negative, while the condition \(y^3>0\) implies that y is positive since the cube of any positive number remains positive.

Step 2: Map the conditions to quadrants

Next move to the Cartesian coordinate system. In the 2D Cartesian plane, it is divided into 4 quadrants. The first quadrant (top right) is where both x and y are positive. The second quadrant (top left) is where x is negative and y is positive. The third quadrant (bottom left) is where both x and y are negative. And the last, the fourth quadrant (bottom right) is where x is positive and y is negative. Now, according to our conditions - x is negative and y is positive, these conditions are met in the second quadrant.

Step 3: Identify the quadrant

Therefore, the only quadrant satisfying both conditions (\(x^3<0\) and \(y^3>0\)) is the second quadrant.