Graphs and Systems of Linear Inequalities - Video Notes

Vocabulary flashcards covering key concepts from solving and graphing linear inequalities and systems, including boundary behavior, shading, and feasible regions.

System of linear inequalities

Two or more inequalities in the same variables; the solution set is the intersection of their individual solution regions (the feasible region).

Inequality

A relation using >,

Is the point (1, 5) a solution to y \ge -2x + 4?

Substitute the point: 5 \ge -2(1) + 4 simplifies to 5 \ge 2. Since this is true, (1, 5) is a solution. If it were false, it would not be a solution.

Graph of an equation

The set of all ordered pairs (x, y) that satisfy the equation; for a linear equation, this is a straight line.

Boundary line

The line obtained by replacing the inequality with its equality (e.g., y = -2x + 4); it separates the solution region from non-solution regions.

Solid line

Indicates the boundary is included in the solution (\ge or \le).

Dashed line

Indicates the boundary is not included in the solution ( > or < ).

Shaded region

The region of the plane that satisfies the inequality; shading shows where the inequality is true.

Graph the inequality y > -2x + 4

The boundary line is y = -2x + 4. It is a dashed line with a y-intercept at (0, 4) and an x-intercept at (2, 0). Shade the region above the line.

For the inequality 3x + 2y \le 6, describe the boundary line and shading.

The boundary line is 3x + 2y = 6. It is a solid line (due to \le). To find shading, test point (0,0): 3(0) + 2(0) \le 6 \implies 0 \le 6. Since this is true, shade the region containing (0,0) (below or to the left of the line).

Feasible region

The common shaded region that satisfies all inequalities in a system.

Corner point (vertex)

A vertex of the feasible region; typically the intersection of two constraint boundaries.

Intersection point

The point where two or more lines cross; in a system, it is a potential solution to the equations.