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Magazine Chapter 3: Problem 7
Explain how you can tell if the region described by the inequality \(3 x-5 y<15\) is above or below the boundary line of \(3 x-5 y=15\)
Short Answer
Expert verified
The region \(3x - 5y < 15\) is below the boundary line \(3x - 5y = 15\).
Step by step solution
01
- Understand the Boundary Line
First, recognize that the inequality shares the same form as the equation of the boundary line, which is given by the equality: \(3x - 5y = 15\). This is a linear equation, and it represents a straight line on the coordinate plane.
02
- Graph the Boundary Line
Graph the line \(3x - 5y = 15\). To graph this line, find two points that satisfy the equation. For instance, when \(x = 0\), \(y = -3\); and when \(y = 0\), \(x = 5\). Plot these points (0, -3) and (5, 0), then draw a straight line through them.
03
- Choose a Test Point
Choose a test point that is not on the boundary line to determine which side of the line represents the inequality. A convenient choice is the origin, (0, 0), but make sure it does not lie on the line you just graphed.
04
- Substitute the Test Point
Substitute the coordinates of the test point (0, 0) into the inequality \(3x - 5y < 15\). This gives: \(3(0) - 5(0) < 15\), which simplifies to \(0 < 15\). This is a true statement.
05
- Interpret the Result
Since the test point (0, 0) satisfies the inequality, it means that the side of the boundary line where (0, 0) is located satisfies the inequality \(3x - 5y < 15\). Therefore, the region containing the origin is the solution region.
06
- Conclude the Position
Since the origin (0, 0) lies below the line \(3x - 5y = 15\), the region described by the inequality \(3x - 5y < 15\) is below the boundary line.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
linear equations
A linear equation is a mathematical statement that shows the relationship between two variables, usually represented as x and y. These relationships can be written in the form of: \[Ax + By = C\].
In its simplest form, it represents a straight line on a graph.
For example, the equation in our exercise is: \[3x - 5y = 15\].
This equation tells us that every pair of (x, y) coordinates that satisfy this equation will lie on a straight line.
Understanding linear equations is crucial, as they form the basis for many problems involving graphing inequalities.
In its simplest form, it represents a straight line on a graph.
For example, the equation in our exercise is: \[3x - 5y = 15\].
This equation tells us that every pair of (x, y) coordinates that satisfy this equation will lie on a straight line.
Understanding linear equations is crucial, as they form the basis for many problems involving graphing inequalities.
coordinate plane
The coordinate plane, also known as a Cartesian plane, is a two-dimensional space defined by two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
The point where these axes intersect is called the origin, represented by (0, 0).
Points on the plane are usually labeled as (x, y), where x indicates the horizontal position and y indicates the vertical position.
In our exercise:
The point where these axes intersect is called the origin, represented by (0, 0).
Points on the plane are usually labeled as (x, y), where x indicates the horizontal position and y indicates the vertical position.
In our exercise:
- We used the coordinate plane to graph the boundary line described by the equation \[3x - 5y = 15\].
- We identified points on this line, such as (0, -3) and (5, 0), to accurately draw it.
- The line divides the plane into two distinct regions. Understanding the coordinate plane helps us visualize and solve the inequality graphically.
test point method
The test point method is a technique used to determine which region of the coordinate plane satisfies a given inequality. Here's how it works:
- First, graph the boundary line of the inequality. In our exercise, this was \[3x - 5y = 15\].
- Next, choose a test point that is not on the boundary line. A common choice is the origin (0, 0), as it's easy to work with.
- Substitute the coordinates of the test point into the inequality. For \[3x - 5y < 15\], substituting (0, 0) gives \[3(0) - 5(0) < 15\], which simplifies to \[0 < 15\], a true statement.
- The result tells us which side of the boundary line is part of the solution region. If the test point satisfies the inequality, then the region containing the test point is the solution region. Simple and effective!
solution region
The solution region of an inequality is the area on the coordinate plane where all points satisfy the inequality. This region is often either above or below the boundary line:
- First, graph the boundary line. For \[3x - 5y < 15\], the boundary line is \[3x - 5y = 15\].
- Next, use the test point method to determine which side of the line is the solution region. In our example, substituting the origin (0, 0) into the inequality gave a true statement, so the region containing (0, 0) is the solution.
- Finally, since (0, 0) is below the boundary line \[3x - 5y = 15\], the region described by the inequality \[3x - 5y < 15\] is below this line.
Remember, shading the solution region visually helps understand and solve inequalities.
