Which Is The Graph Of Linear Inequality 2Y > X - 2?
Linear inequalities are equations that use greater than, less than, greater than or equal to, or less than or equal to signs to compare two expressions. In this article, we will focus on the linear inequality 2y > x - 2 and explore its graph. Understanding the graph of this inequality is essential in solving problems that involve linear inequalities.
What is a Linear Inequality?
A linear inequality is an equation that describes a relationship between two variables using inequality symbols such as > or <. In other words, it is a mathematical statement that one expression is greater or less than another expression. Linear inequalities are used to represent situations where there are multiple possible solutions, such as when a company produces a range of products or when a person earns a salary within a certain range.
Linear Inequality 2y > x - 2
Linear inequality 2y > x - 2 is a mathematical expression that compares the value of 2 multiplied by y to the value of x minus 2. This inequality can be written in different forms, such as y > (1/2)x - 1 or x - 2y < -2. These forms may look different, but they all represent the same relationship between x and y.
Graphing Linear Inequality 2y > x - 2
To graph the linear inequality 2y > x - 2, we need to plot the line y = (1/2)x - 1, which is the boundary line of the inequality. This line separates the plane into two regions: the region above the line and the region below the line. To determine which region satisfies the inequality, we choose a test point that is not on the line and substitute its x and y values into the inequality. If the inequality is true, then the region that contains the test point is the solution.
For example, let's choose the test point (0,0). Substituting x=0 and y=0 into the inequality, we get:
Since 0 is greater than -2, the point (0,0) is in the region above the line y = (1/2)x - 1. Therefore, the solution to the inequality 2y > x - 2 is the region above the line.
Shading the Solution
To shade the solution to the inequality 2y > x - 2, we need to shade the region above the line y = (1/2)x - 1. This region includes all the points where 2y is greater than x - 2. We can use a test point to check whether a point is in the shaded region or not. If the point satisfies the inequality, it is in the shaded region. If it does not satisfy the inequality, it is not in the shaded region.
For example, let's choose the point (2,3). Substituting x=2 and y=3 into the inequality, we get:
Since 6 is greater than 0, the point (2,3) satisfies the inequality 2y > x - 2 and is in the shaded region above the line y = (1/2)x - 1.
Interpreting the Graph
The graph of the linear inequality 2y > x - 2 shows all the possible solutions to the inequality. In this case, the shaded region above the line y = (1/2)x - 1 represents all the points that satisfy the inequality. Any point in this region makes 2y greater than x - 2. The boundary line y = (1/2)x - 1 is not part of the solution because the inequality is strict. That is, 2y is greater than x - 2, not equal to it.
Applications of Linear Inequalities
Linear inequalities are used in a variety of fields, including economics, engineering, and science. In economics, linear inequalities are used to model constraints such as limited resources, production capacity, and demand. In engineering, linear inequalities are used to model constraints such as strength, weight, and safety. In science, linear inequalities are used to model constraints such as time, space, and energy.
Example Problem
Suppose a company produces two types of products: product A and product B. The production process for product A requires 3 hours of machine time and 2 hours of labor, while the production process for product B requires 2 hours of machine time and 4 hours of labor. The company has a total of 48 hours of machine time and 32 hours of labor available per week. Let x be the number of units of product A produced and y be the number of units of product B produced. Write a system of linear inequalities to model the production process and graph the solution.
The system of linear inequalities is:
To graph the solution, we first graph the boundary lines of the inequalities:
Then, we shade the region that satisfies all the inequalities:
The shaded region represents all the possible combinations of units of product A and product B that the company can produce given the constraints. For example, the company can produce 16 units of product A and 4 units of product B or 12 units of product A and 5 units of product B, among other combinations.
Conclusion
In summary, the graph of the linear inequality 2y > x - 2 is the region above the line y = (1/2)x - 1. This region includes all the points where 2y is greater than x - 2. Linear inequalities are used to model situations where there are multiple possible solutions, and they are applied in various fields such as economics, engineering, and science. Understanding how to graph linear inequalities is essential in solving problems that involve constraints and optimization.
So, if you are dealing with a linear inequality problem, don't forget to graph it to visualize the solution!
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