📊 Linear Inequalities: Inequality Symbols & Number Line

1. Comparison Relations & Verbal Meanings

• Strictly Less Than ($<$): The value is strictly smaller than the boundary. Verbal cues: under, less than, below.
• Strictly Greater Than ($>$): The value is strictly larger than the boundary. Verbal cues: more than, greater than, exceeds, above.
• Less Than or Equal To ($\le$): The value is at most the boundary. Verbal cues: at most, maximum of, no more than.
• Greater Than or Equal To ($\ge$): The value is at least the boundary. Verbal cues: at least, minimum of, no less than.
• Not Equal To ($\neq$): Any real number except the boundary point.

2. Number Line Graphing Rules

• Boundary Point Circle Type:
  - Open Circle ($\circ$, Hollow): Used for strict inequalities ($<$ and $>$). The boundary value itself is not part of the solution.
  - Solid Circle ($\bullet$, Filled): Used for inclusive inequalities ($\le$ and $\ge$). The boundary value is part of the solution.
  
• Ray Shading Direction:
  - Rightward Shading ($>$ or $\ge$): Shading extends to the right towards positive infinity ($+\infty$).
  - Leftward Shading ($<$ or $\le$): Shading extends to the left towards negative infinity ($-\infty$).
  - Not Equal To ($\neq$): An open circle ($\circ$) is placed at the boundary point, and the line is shaded in both directions ($< b$ and $> b$).

• Step 1 — Identify the Boundary Point: Locate the numerical value where the condition changes.
• Step 2 — Determine Circle Style: Inspect the inequality symbol. Choose an open circle ($\circ$) for strict ($<$, $>$) or a solid circle ($\bullet$) for inclusive ($\le$, $\ge$).
• Step 3 — Determine Ray Direction: Point the arrow right for larger values ($>$ or $\ge$) or left for smaller values ($<$ or $\le$).
• Step 4 — Verify with a Test Point: Select a simple test value (such as $x = 0$) in the shaded region and verify that substituting it makes the inequality true.