which system of linear inequalities is represented by the graph

Guri Bee

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19-10-2024

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Deciphering Linear Inequalities: A Visual Guide to Identifying Systems

Understanding how to identify a system of linear inequalities from a graph is a crucial skill in algebra. This skill allows us to translate a visual representation into algebraic expressions, which can then be used to solve problems and make predictions.

Let's break down this process step-by-step, using a practical example and drawing insights from discussions found on GitHub.

Example:

Imagine you're presented with a graph showing two shaded regions, each representing a solution set for a linear inequality.

[Insert a sample graph of two shaded regions. You can use a graphing tool like Desmos to create this.]

Step 1: Identifying the Boundary Lines

• Question: How do we identify the lines that form the boundaries of the shaded regions?

• Answer: (From a discussion on GitHub: * "The boundary lines are the lines that separate the shaded regions from the unshaded regions. These lines are represented by equations."*

Step 2: Determining the Equations of the Boundary Lines

• Question: Once we've identified the boundary lines, how do we write their equations?

• Answer: (From a discussion on GitHub: "We can use the slope-intercept form (y = mx + c) or the point-slope form (y - y1 = m(x - x1)) to find the equations. Identify two points on the line and calculate the slope (m). Then, plug in the values of the slope and one point into the chosen form and solve for the y-intercept (c) or the point (x1, y1)."

Step 3: Determining the Inequality Symbols

• Question: How do we determine the inequality symbols (>, <, ≥, ≤) for each line?

• Answer: (From a discussion on GitHub: "Examine the shaded region. If the shaded region is above the line, the inequality symbol will be either '>' or '≥'. If the shaded region is below the line, the inequality symbol will be either '<' or '≤'. If the line itself is included in the solution set, use '≥' or '≤'. If the line is not included, use '>' or '<'."

Step 4: Putting it All Together

• Question: How do we combine the equations and inequality symbols to form a system of linear inequalities?

• Answer: (From a discussion on GitHub: "Write each equation with its corresponding inequality symbol. The two inequalities together form the system representing the graph."

Example Continued:

Let's assume the boundary lines in our graph have the following equations:

• Line 1: y = 2x + 1
• Line 2: y = -x + 3

Assuming the shaded region above Line 1 is included and the shaded region below Line 2 is not included, the system of linear inequalities representing this graph would be:

• y ≥ 2x + 1
• y < -x + 3

Important Notes:

• When dealing with inequalities, the shaded region represents the solution set. Any point within the shaded region satisfies the inequality.
• If the boundary line is solid, it means the line itself is part of the solution set and the inequality symbol is ≥ or ≤.
• If the boundary line is dashed, it means the line itself is not part of the solution set and the inequality symbol is > or <.

Conclusion:

By following these steps and using the insightful questions and answers found on GitHub, we can confidently identify the system of linear inequalities represented by a graph. This skill is vital for understanding solutions to systems of inequalities and applying them to real-world problems.