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Magazine Chapter 3: Problem 108
Graph the solution set. $$ y>0 $$
Short Answer
Expert verified
Shade the region above the x-axis; draw a dashed line along the x-axis.
Step by step solution
01
Understand the Inequality
The given inequality is \( y > 0 \), which means we are looking for all the points on a coordinate plane where the y-coordinate is greater than zero.
02
Describe the Region
The y-values greater than 0 are located above the x-axis on a coordinate plane. Therefore, the solution set is the entire plane above (but not including) the x-axis.
03
Determine the Boundary
The boundary for our inequality \( y > 0 \) is the x-axis itself where \( y = 0 \). However, since \( y \) must be greater than 0, the x-axis will be a dashed line to indicate that it is not included in the solution set.
04
Represent on the Graph
To graph the solution, draw a horizontal dashed line along the x-axis to show \( y = 0 \). Shade the region above the x-axis where \( y > 0 \) to represent the solution set.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
graphing inequalities
Graphing inequalities involves representing solutions of inequalities visually on a graph. It helps in understanding which regions describe the solutions to a given inequality. For example, when graphing the inequality \( y > 0 \), you begin by defining the boundary, which is the line \( y = 0 \) here.
- This line serves as the base to visualize where \( y \) can be greater than zero.
- In graphing, it's important to decide whether to use a solid line or a dashed line for the boundary. A solid line is used when the points on the line are part of the solution, and a dashed line is used when they aren't.
coordinate plane
The coordinate plane is an essential tool for graphing linear inequalities. It consists of two perpendicular axes, the x-axis, and the y-axis, which intersect at the origin (0,0).
- The plane is divided into four quadrants, where each point is defined by an ordered pair \((x, y)\).
- This visualization enables the representation of algebraic expressions in geometric terms, like graphing inequalities.
solution set
In the context of inequalities, a solution set refers to all points on a graph that satisfy the inequality condition. For the inequality \( y > 0 \), the solution set comprises all points with a y-coordinate greater than zero.
- This means any point located above the x-axis is part of the solution set.
- Points on the x-axis itself are not included since \( y \) must be strictly greater than zero.
boundary line
The boundary line is crucial when graphing inequalities as it defines the limit where the inequality changes. For the inequality \( y > 0 \), the boundary line is given by the equation \( y = 0 \), which coincides with the x-axis.
- In regions where \( y \) needs to be greater than zero, the boundary indicates where values transition from satisfying to not satisfying the inequality.
- Since \( y > 0 \) excludes the line itself, you represent it using a dashed line to show it's not included.
