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Chapter 3: Problem 31

Explain why the graph of \(y>x\) cannot lie in quadrant IV.

Short Answer

Expert verified
The graph of y > x cannot lie in Quadrant IV because in that quadrant y-coordinates are negative and x-coordinates are positive, making it impossible for y to be greater than x.

Step by step solution

01

Recall the definition of Quadrant IV

In the Cartesian coordinate system, Quadrant IV is the region where the x-coordinates are positive and the y-coordinates are negative.
02

Understand the inequality y > x

The inequality y > x means that for any point (x, y) on this graph, the y-coordinate must be greater than the x-coordinate.
03

Analyze the conditions in Quadrant IV

In Quadrant IV, all x-coordinates are positive (x > 0) and all y-coordinates are negative (y < 0). Since y < 0 and x > 0 in Quadrant IV, it is impossible for y to be greater than x.
04

Conclusion

Since no points in Quadrant IV can satisfy the inequality y > x, the graph of y > x cannot lie in Quadrant IV.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cartesian coordinate system
The Cartesian coordinate system is a method for describing a position in a plane using two numbers, usually denoted as \((x, y)\). This system divides the plane into four quadrants by intersecting two perpendicular lines called axes.
The horizontal line is the x-axis, and the vertical line is the y-axis. The point where these axes intersect is known as the origin, labeled as \((0, 0)\).
Each quadrant has specific characteristics:
  • Quadrant I: Both x and y coordinates are positive \((x > 0, y > 0)\).
  • Quadrant II: x is negative and y is positive \((x < 0, y > 0)\).
  • Quadrant III: Both x and y coordinates are negative \((x < 0, y < 0)\).
  • Quadrant IV: x is positive and y is negative \((x > 0, y < 0)\).
By understanding these properties, you can easily determine the position of a point in the Cartesian plane.
inequality y > x
The inequality \(y > x\) represents all the points \((x, y)\) where the y-coordinate is greater than the x-coordinate. This inequality helps define a specific region in the Cartesian coordinate system.
To understand \(y > x\), imagine a line where every point has coordinates such that y equals x. This line is written as \(y = x\).
Points that satisfy \(y > x\) lie above this line because the y-coordinate at these points is greater than the corresponding x-coordinate. For example, the point \((2, 3)\) satisfies \(y > x\) because 3 is greater than 2.
Understanding this inequality helps identify and graph the region it delineates in the coordinate system.
Quadrant IV analysis
Quadrant IV in the Cartesian coordinate system is the bottom-right region. In this area, x-coordinates are positive \((x > 0)\) while y-coordinates are negative \((y < 0)\).
Because of this, any point in Quadrant IV will have a positive x-value and a negative y-value. This unique characteristic informs our understanding of inequalities within this region.
When analyzing the inequality \(y > x\) in Quadrant IV, we see that it's impossible for the inequality to hold. Since y-values are always negative and x-values are positive, the y-coordinate \((y < 0)\) can never be greater than the x-coordinate \((x > 0)\).
Thus, no points in Quadrant IV satisfy the inequality \(y > x\).
graphing inequalities
Graphing inequalities involves shading a region on the Cartesian plane where the inequality holds true. For \(y > x\), we graph the line \(y = x\) first.
To represent the inequality, we shade the area above this line because any point above it will have a y-coordinate greater than the x-coordinate.
Tools to remember when graphing inequalities:
  • Use a solid line for \(鈮) or \(鈮), and a dashed line for \(>\) or \(<\).
  • Shade the correct region of the graph where the inequality is true.
Applying these steps, you can graphically visualize the solution to an inequality, making it easier to interpret and analyze its meaning within the coordinate system.

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