which compound inequality is represented by the graph

Guri Bee

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21-10-2024

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Decoding Inequalities: Understanding Graphs and their Compound Counterparts

Compound inequalities are mathematical expressions that combine two or more inequalities using the words "and" or "or". They are often represented visually through number lines, making it crucial to understand how to translate these graphical representations into their corresponding algebraic forms.

Let's delve into the process of identifying the compound inequality represented by a graph, using examples and insights from GitHub discussions.

Identifying the Inequality Type: "And" or "Or"

The first step in deciphering a graph is to determine whether the compound inequality uses "and" or "or."

"And" Inequalities: These represent the intersection of two inequalities. The solution is the set of values that satisfy both inequalities simultaneously. Graphically, this is depicted as a shaded area on the number line where the solutions of both individual inequalities overlap.

"Or" Inequalities: These represent the union of two inequalities. The solution includes values that satisfy at least one of the inequalities. On a number line, this means the shaded area encompasses all the solutions of each individual inequality.

Example from GitHub:

Let's examine a real-world example. In a GitHub discussion https://github.com/openai/whisper/issues/2414, a user sought clarification on interpreting a graph.

The graph: Imagine a number line where the region between -2 and 5 is shaded, including the points -2 and 5. This indicates that the solution set includes all values greater than or equal to -2 and less than or equal to 5.

The solution: This graph represents the compound inequality "x ≥ -2 and x ≤ 5". It implies that the solution includes all values that satisfy both conditions.

Key Considerations for Graph Analysis:

• Open vs. Closed Circles: A closed circle on the number line indicates that the endpoint is included in the solution set (≥ or ≤). An open circle means the endpoint is excluded (< or >).
• Direction of Shading: Shading to the right represents values greater than the endpoint, while shading to the left indicates values less than the endpoint.

Practical Application:

Compound inequalities often arise in real-world scenarios. For example, imagine a store offering a discount on purchases between $20 and $50. The graph representing this scenario would have closed circles at 20 and 50, with the shaded region encompassing all values between them. This visual representation clearly depicts the discount range.

Conclusion:

Understanding how to interpret graphs of compound inequalities is crucial for translating visual information into mathematical expressions. By recognizing the key elements – "and" or "or", open or closed circles, and shading direction – we can accurately represent the solution set in algebraic form. With practice and attention to detail, deciphering these graphs becomes a straightforward process.