To determine in which quadrant(s) an ordered pair with opposite signs must lie, we first need to understand the layout of the coordinate plane.
The coordinate plane is a two-dimensional plane formed by two perpendicular number lines called the x-axis and the y-axis. The intersection of these two lines is called the origin, denoted by the point (0, 0). The x-axis represents the horizontal direction, with positive values to the right of the origin and negative values to the left. The y-axis represents the vertical direction, with positive values above the origin and negative values below.
The coordinate plane is divided into four quadrants: Quadrant I, Quadrant II, Quadrant III, and Quadrant IV.
- Quadrant I: Consists of all the points with both x and y coordinates positive.
- Quadrant II: Consists of all the points with x coordinates negative and y coordinates positive.
- Quadrant III: Consists of all the points with both x and y coordinates negative.
- Quadrant IV: Consists of all the points with x coordinates positive and y coordinates negative.
Now, if an ordered pair has opposite signs, it means one coordinate is positive while the other is negative.
For example, let's consider an ordered pair (-3, 4). The x-coordinate is negative (-3), indicating that it falls to the left of the origin. The y-coordinate is positive (4), indicating that it falls above the origin. Hence, this ordered pair lies in Quadrant II.
In general, if the x-coordinate is negative while the y-coordinate is positive, the ordered pair lies in Quadrant II. Similarly, if the x-coordinate is positive while the y-coordinate is negative, the ordered pair lies in Quadrant IV.
Therefore, an ordered pair with opposite signs must lie in either Quadrant II or Quadrant IV.