Overview
This lecture explains how to solve and graph systems of linear inequalities, focusing on shading regions that satisfy all given conditions.
Graphing Linear Inequalities
- Graph each inequality as a line: solid for ≤ or ≥, dashed for < or >.
- For x ≥ a, draw a vertical line at x = a and shade to the right.
- For y < b, draw a horizontal dashed line at y = b and shade below.
- The solution region is where the shading for all inequalities overlaps.
Graphing Inequalities with Non-horizontal/Vertical Lines
- Write linear inequalities in slope-intercept form (y = mx + b) for easier graphing.
- For y > mx + b, shade above the line; for y < mx + b, shade below.
- When comparing two lines, the solution is where the correct shaded regions overlap.
Working with Multiple Inequalities
- Restate equations in y = mx + b form for clarity.
- For example, 2x + 3y > 6 becomes y > (-2/3)x + 2.
- Graph each line, using the appropriate solid or dashed style.
- Identify and shade the region that satisfies all inequalities simultaneously.
Example Approaches
- For three inequalities (e.g., x > 1, y ≤ 6, y > 3/2x - 3), graph each and shade the region where all three overlap.
- With standard form equations, solve for y before graphing to find slope and intercept.
Key Terms & Definitions
- System of Linear Inequalities — Two or more linear inequalities considered at the same time.
- Shaded Region — Area where all inequalities in the system are true.
- Slope-Intercept Form — Equation of a line written as y = mx + b, where m is the slope and b is the y-intercept.
- Dashed Line — Used for strict inequalities (< or >); points on the line are not included.
- Solid Line — Used for inclusive inequalities (≤ or ≥); points on the line are included.
Action Items / Next Steps
- Practice graphing systems of inequalities and identifying the correct solution region.
- Convert standard form equations to slope-intercept form before graphing.
- Prepare for any related homework or quizzes on graphing inequalities.