write a system of linear inequalities represented by the graph

Guri Bee

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21-10-2024

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Deciphering the Code: Writing Systems of Linear Inequalities from Graphs

Have you ever looked at a graph shaded with different colors and wondered what inequalities it represents? Understanding how to write a system of linear inequalities from a graph is a crucial skill in algebra, allowing you to translate visual information into symbolic representations.

Let's break down the process step-by-step, drawing upon insights from the GitHub repository (replace with the actual repository link), which provides valuable resources for learning about linear inequalities.

Key Components

Before we dive into the process, let's recall some fundamental concepts:

• Linear Inequality: A mathematical statement comparing two expressions using inequality symbols (>, <, ≥, ≤).
• System of Inequalities: A set of two or more linear inequalities, representing a region in a coordinate plane where all the inequalities are satisfied simultaneously.
• Boundary Line: The line that defines the edge of the shaded solution region. It can be solid (inclusive of the line) or dashed (exclusive).

Decoding the Graph

1. Identify the Boundary Lines: Each shaded region is defined by one or more boundary lines. Pay attention to whether the lines are solid or dashed.

• Solid Line: Indicates the inequality includes the line itself, represented by "≥" or "≤".
• Dashed Line: Indicates the inequality excludes the line, represented by ">" or "<".

2. Determine the Slope and Y-intercept of each line: Remember, the equation of a line can be written in slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept.

3. Choose a Test Point: Select a point not on any boundary line. This point will help you determine the direction of the inequality.

4. Substitute the Test Point into the Equation: Substitute the coordinates of the test point into the equation of the line.

• True Statement: If the resulting statement is true, the shaded region contains the test point, and the inequality points towards the shaded region.
• False Statement: If the resulting statement is false, the shaded region does not contain the test point, and the inequality points away from the shaded region.

5. Write the Inequality: Combine the slope-intercept form with the appropriate inequality symbol based on the test point result and the line type (solid or dashed).

Example:

Let's consider a graph with a shaded region above a solid line passing through (0, 2) with a slope of 1.

1. Boundary Line: Solid line implies "≥" or "≤".
2. Slope and Y-intercept: m = 1, b = 2. Equation: y = x + 2
3. Test Point: Let's choose (0, 3).
4. Substitution: 3 ≥ 0 + 2 (True)
5. Inequality: y ≥ x + 2

Additional Considerations:

• Multiple Shaded Regions: Each shaded region in a multi-region graph corresponds to a different inequality in the system.
• Special Cases: If the boundary line is vertical (x = a), the inequality will be x > a or x < a. Similarly, for a horizontal line (y = b), the inequality will be y > b or y < b.

Practice Makes Perfect:

The best way to master this skill is to practice! The GitHub repository offers a variety of graph examples and practice exercises, allowing you to test your understanding and build your confidence.

Conclusion:

Writing systems of linear inequalities from graphs involves a systematic approach of analyzing the boundary lines, determining the slope and y-intercept, choosing a test point, and finally writing the corresponding inequalities. By understanding the process and applying the concepts explained above, you can confidently translate visual representations of inequalities into symbolic expressions, unlocking a deeper understanding of this fundamental concept in algebra.