Graph the solution to the inequality 4x+5y 20 – Graph the solution to the inequality 4x+5y ≤ 20 and step into the captivating realm of linear inequalities. This guide will illuminate the intricacies of graphing inequalities, empowering you to visualize solutions and make informed decisions.
Prepare to embark on a journey of discovery, where we unravel the concept of linear inequalities, explore the significance of boundary lines, and delve into the art of shading solution regions. Along the way, you’ll uncover the practical applications of graphing inequalities in diverse fields, solidifying your understanding of this fundamental mathematical concept.
Graphing the Solution to the Inequality: Graph The Solution To The Inequality 4x+5y 20
Graphing the solution to an inequality involves plotting the boundary line and shading the correct region that satisfies the inequality. Linear inequalities are represented graphically as lines, and the solution region is the area on one side of the line.
Identifying the Boundary Line, Graph the solution to the inequality 4x+5y 20
To find the intercepts of the inequality 4x+5y ≤ 20, set x = 0 and solve for y, and vice versa. This gives us the intercepts (0, 4) and (5, 0).
The boundary line is the line passing through these intercepts. The inequality symbol ≤ indicates that the line is included in the solution region, so the boundary line is a solid line.
Shading the Solution Region
To determine which side of the boundary line to shade, choose a test point that is not on the line. If the inequality holds true for the test point, then shade the region containing the test point.
For example, the test point (0, 0) satisfies the inequality 4x+5y ≤ 20, so we shade the region below the boundary line.
| Test Point | Value | Conclusion |
|---|---|---|
| (0, 0) | 4(0) + 5(0) = 0 ≤ 20 | Shade below the boundary line |
| (1, 1) | 4(1) + 5(1) = 9 ≤ 20 | Shade below the boundary line |
Applications of the Graph
Graphing inequalities has practical applications in various fields:
- Economics:Inequalities can represent budget constraints, production possibilities, and supply and demand curves.
- Engineering:Inequalities can be used to design structures, optimize processes, and analyze data.
- Optimization:Inequalities can help determine the maximum or minimum values of functions within specified constraints.
Essential FAQs
What is the significance of the boundary line in graphing inequalities?
The boundary line represents the equation of the inequality (4x+5y = 20 in this case), dividing the coordinate plane into two regions: one where the inequality is true and one where it is false.
How do I determine which region to shade in when graphing an inequality?
Choose a test point not on the boundary line and substitute its coordinates into the inequality. If the inequality holds true, shade the region containing the test point. If it does not hold true, shade the opposite region.
What are some practical applications of graphing inequalities?
Graphing inequalities finds applications in various fields, including economics (budget constraints), engineering (design constraints), and optimization (maximizing or minimizing functions).
