Graphing Step 1: Using Open vs. Closed Circles for the Boundary Point
When graphing inequalities on a number line, the boundary point is marked with either a closed circle or an open circle to indicate inclusion or exclusion — a foundational graphing skill in enVision Algebra 1 Chapter 1 for Grade 11. A closed circle (●) means the boundary is included in the solution set and is used with ≤ or ≥. An open circle (○) means the boundary is excluded and is used with < or >. For x ≥ 3, place a closed circle at 3 and shade right. For x < -2, place an open circle at -2 and shade left. Getting the circle type right is the first critical step before shading.
Key Concepts
Use a closed circle (●) for inclusive inequalities ($\leq$ or $\geq$) and an open circle (○) for strict inequalities ($<$ or $ $). The circle indicates whether the boundary point is included in the solution set.
Common Questions
When do you use a closed circle on a number line?
Use a closed circle (●) when the inequality uses ≤ or ≥, meaning the boundary value is included in the solution. For example, x ≥ 3 gets a closed circle at 3.
When do you use an open circle on a number line?
Use an open circle (○) when the inequality uses < or >, meaning the boundary value is not included. For example, x < -2 gets an open circle at -2.
For x ≤ 0, which circle and which direction?
Closed circle at 0 (because ≤ includes the boundary) and shade left (because x is less than or equal to 0).
What does the circle type tell you about the solution set?
A closed circle means the boundary point satisfies the inequality and is part of the solution. An open circle means the boundary point does not satisfy it and is excluded.
How does this relate to piecewise functions?
Piecewise functions use the same notation: closed circles at boundary points where the function equals that value, and open circles where the boundary is approached but not reached.