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Chapter 4: Problem 37

Construct a system of linear inequalities that describes all points in the first quadrant.

Short Answer

Expert verified
The system of inequalities is \( x > 0 \) and \( y > 0 \).

Step by step solution

01

Understand the First Quadrant

The first quadrant of a Cartesian coordinate system is the region where both the x-values and y-values are positive. In other words, in this region, both x and y are greater than zero.
02

Define Inequalities for x and y

To describe the first quadrant with inequalities, we need the conditions that maintain x and y in the positive region. The appropriate inequalities are: \( x > 0 \) and \( y > 0 \). These inequalities ensure both x and y remain positive.
03

Include Boundary Conditions

Sometimes, to include the boundary lines (the axes themselves) in the region, we can modify the inequalities by allowing zero. This is done by writing \( x \geq 0 \) and \( y \geq 0 \). This includes all points on the axes in addition to the region where x and y are both positive. However, just specifying \( x > 0 \) and \( y > 0 \) suffices since the question asks for the first quadrant, typically considered excluding the axes.

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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 way of identifying the location of points in a plane using two perpendicular lines. These lines are called axes. The horizontal line is known as the x-axis, while the vertical line is called the y-axis. Every point can be referred to by an ordered pair of numbers:
  • The first number in the pair refers to the horizontal position along the x-axis.
  • The second number represents the vertical position along the y-axis.
These numbers are called coordinates. They tell us exactly where a point is positioned on the plane. The point where the x-axis and y-axis meet is called the origin, which has coordinates (0, 0). The Cartesian coordinate system divides the plane into four sections, termed as quadrants. Each quadrant contains points with distinct combinations of positive and negative coordinates.
first quadrant
The first quadrant is one of the four quadrants formed by the Cartesian coordinate system. In this section of the plane, both the coordinates \(x\) and \(y\) are positive. It is essentially the top right corner when visualizing the graph. This means all the points in this quadrant will have positive x-values and y-values.Some key points to remember about the first quadrant:
  • It includes points like (2, 3), (5, 10), etc. where both numbers in the pair are positive.
  • This is the only quadrant where you will find pairs of positive values of both coordinates.
  • Equations representing the first quadrant are typically linear inequalities like \(x > 0\) and \(y > 0\).
boundary conditions
Boundary conditions refer to whether or not the borders (axes, in this case) are included in the solution set. In the exercise we discussed, the inequalities are used to distinguish whether the boundary lines are part of the solution.To include boundaries of the first quadrant:
  • If you want to exclude the axes, the standard inequalities are \(x > 0\) and \(y > 0\).
  • Conversely, to incorporate the axes, the conditions become \(x \geq 0\) and \(y \geq 0\). This means that any point lying exactly on the x-axis or y-axis is also considered as part of the first quadrant.
Including or excluding boundary points is crucial depending on the problem's requirements. In many cases in geometry and algebra, this delineation helps define very clear and specific regions on the plane.

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