Graphing Linear Inequalities - Algebra-1

1. Fundamental Concepts

Graphing linear inequalities is the process of visually representing the solution set of a linear inequality on a coordinate plane. For a two-variable linear inequality (e.g., \(y \gt mx + b\)), the solution set is not a single point or line but a region consisting of all points \((x, y)\) that satisfy the inequality. This region is divided by the corresponding linear equation (called the "boundary line"), and the solution set is determined by both the nature of the boundary line (solid or dashed) and the position of the region (on one side of the boundary line).

Boundary Line: The equation form of the linear inequality (e.g., the boundary line of \(y \leq 2x + 3\) is \(y = 2x + 3\)), which divides the coordinate plane into regions.
- If the inequality contains "\(\leq\)" or "\(\geq\)", the boundary line is solid, indicating that points on the line are part of the solution set;
- If the inequality contains "<" or ">", the boundary line is dashed, indicating that points on the line are not part of the solution set.
Solution Region: The area on one side of the boundary line where all points \((x, y)\) satisfy the inequality. A "test point" is used to determine the position of this region (usually the origin \((0,0)\); if the origin lies on the boundary line, another point like \((1,0)\) is chosen).
Difference Between Single-Variable and Two-Variable Inequalities:
- A single-variable linear inequality (e.g., x > 5) is represented as a ray on a number line or a region on one side of a vertical/horizontal line in a coordinate plane;
- A two-variable linear inequality (e.g., y < -x + 2) is represented as a region on one side of an inclined line in a coordinate plane.

Easy: Graph \(x \geq -2\) Steps:
1. The boundary line is \(x = -2\) (a vertical line). Since the inequality contains "\(\geq\)", draw a solid line;
2. Choose the test point \((0,0)\). Substituting gives \(0 \geq -2\) (true), so the solution region is all areas to the right of the boundary line (including the line itself).
Medium: Graph \(y \lt 3x - 1\) Steps:
1. The boundary line is \(y = 3x - 1\) (slope = 3, y-intercept = \(-1\)). Since the inequality contains "<", draw a dashed line;
2. Choose the test point \((0,0)\). Substituting gives 0 < -1 (false), so the solution region is all areas below the boundary line (excluding the line itself).
Hard: Graph the system of inequalities \(\begin{cases} y \geq \frac{1}{2}x + 1 \\ y \lt -2x + 4 \\ x \geq 0 \end{cases}\) Steps:
1. Draw the three boundary lines respectively:
  - \(y = \frac{1}{2}x + 1\) (solid line, due to "\(\geq\)");
  - \(y = -2x + 4\) (dashed line, due to "<");
  - \(x = 0\) (solid line, due to "\(\geq\)");
2. Determine the solution region for each inequality:
  - \(y \geq \frac{1}{2}x + 1\): The region above the boundary line;
  - \(y \lt -2x + 4\): The region below the boundary line;
  - \(x \geq 0\): The region to the right of the y-axis;
3. The overlapping area of the three regions is the solution set of the system, which is shaded in the graph.

Identify and Draw the Boundary Line:
- Convert the inequality to slope-intercept form \(y = mx + b\) (or vertical/horizontal line forms like \(x = a\) or \(y = b\)) to clarify the slope and intercept of the boundary line;
- Draw a solid line for "\(\leq\)" or "\(\geq\)" and a dashed line for "<" or ">".
Use a Test Point to Determine the Region:
- Prefer the origin \((0,0)\) (if not on the boundary line). Substitute it into the inequality:
  - If the inequality holds, the solution region is the side where the test point lies;
  - If not, the solution region is the opposite side of the test point.
Graph Systems of Inequalities:
- Draw the solution region for each inequality separately, using different symbols (e.g., slashes, grids) to distinguish them;
- The overlapping region is the solution set of the system. If there is no overlap, the solution set is empty.
Verify Accuracy:
- Pick any point in the shaded solution region and substitute it into the original inequality to check if it holds;
- Ensure the boundary line’s style (solid/dashed) matches the inequality sign (avoid confusing "\(\leq\)" with "<").