Question
43-52 Sketch the region in the \( x y \)-plan 43. \( \{(x, y) \mid x<0\} \)
Ask by Tyler Kelley. in South Africa
Feb 18,2025
Upstudy AI Solution
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Answer
To sketch the region where \( x < 0 \) in the xy-plane:
1. Draw a vertical dashed line at x = 0 (the y-axis).
2. Shade the area to the left of this line.
This shaded area represents all points where the x-coordinate is less than 0.
Solution
The inequality x < 0 represents the set of all points (x, y) in the xy-plane whose x-coordinate is less than 0, regardless of the y-coordinate. In other words, it's the entire half-plane to the left of the y-axis.
To sketch the region:
1. Draw a vertical dashed line along x = 0 (the y-axis) to indicate the boundary, which is not included in the region because the inequality is strict.
2. Shade the area to the left of this dashed line to represent all points where x is less than 0.
This shaded region is the required solution.
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Simplify this solution Mind Expander
To sketch the region defined by \( \{(x, y) \mid x < 0\} \), start by drawing the y-axis and x-axis on a coordinate plane. The condition \( x < 0 \) means that we only consider points where the x-coordinate is negative. Therefore, the region will be the entire left half of the plane, which includes all points with a negative x-value. It extends infinitely upward and downward from the y-axis, not including the line where \( x = 0 \). Next, you can shade this entire left half of the coordinate plane, clearly showing that all points in this region are valid. Remember, the vertical line along \( x = 0 \) is a boundary but not included in the shaded region. Happy sketching!
