What is Linear Inequality - Algebra-1

1. Fundamental Concepts

A linear inequality is a linear expression containing one or more variables, connected by inequality signs (such as <, >, ≤, ≥). Unlike linear equations (e.g., ), the solution to a linear inequality is not a specific value but a set of all variable values that satisfy the inequality condition, which usually appears as a region in the coordinate system.

Examples:
- Single-variable linear inequality:
- Two-variable linear inequality:

Meaning of inequality signs:
- < (less than) and > (greater than) indicate that the boundary value is not included;
- ≤ (less than or equal to) and ≥ (greater than or equal to) indicate that the boundary value is included.
Boundary line: For a two-variable linear inequality (e.g., $y \gt mx + b$ ), the corresponding linear equation $y = mx + b$ is called the "boundary line."
- If the inequality sign is < or >, the boundary line is a dashed line (points on the boundary are not included);
- If the inequality sign is ≤ or ≥, the boundary line is a solid line (points on the boundary are included).
Solution set region: The solution to a two-variable linear inequality is the region in the coordinate system consisting of all points that satisfy the inequality. The position of the region (above, below, to the left, or to the right of the boundary line) is determined by testing a "test point" (e.g., the origin if it is not on the boundary line).

Easy: Solve the single-variable linear inequality
Solution: Transpose terms to get , divide both sides by 3 to get . The solution set is all real numbers less than 4.
Medium: Graph the two-variable linear inequality Steps:
1. Draw the boundary line (solid line, because of "≥");
2. Take the test point and substitute it into the inequality: (not true), so the solution set is the region above the boundary line.
Hard: Solve the system of inequalities Solution:
1. Draw the boundary lines respectively: (dashed line) and (solid line);
2. Determine the solution set region for each inequality:
  - The solution set of is the region below the dashed line;
  - The solution set of is the region above the solid line;
3. The overlapping part of the two regions is the solution set of the system of inequalities.

Solving single-variable linear inequalities: Similar to solving linear equations, simplify by transposing terms and combining like terms. However, note that: when multiplying or dividing both sides by a negative number, the direction of the inequality sign must be reversed (e.g., simplifies to ).
Graphing two-variable linear inequalities:
- Step 1: Convert the inequality to the slope-intercept form (e.g., ) to determine the slope and intercept of the boundary line;
- Step 2: Draw the boundary line (solid or dashed) according to the type of inequality sign;
- Step 3: Choose a test point not on the boundary line, substitute it into the inequality to determine the solution set region, and shade the region.
Solving systems of inequalities: Solve each inequality separately, and find the overlapping part of all solution set regions in the coordinate system, which is the solution to the system of inequalities.