Graphing Solutions to Systems of Linear Inequalities

Learning Outcomes

  • Practice graphing the solution region for systems of linear inequalities.

  • Understand how to graph solutions to a system involving a compound inequality.

Key Concepts

  • Systems of Linear Inequalities: These are inequalities involving two or more variables where the solutions are represented by regions in the graph.

  • Compound Inequalities: A compound inequality consists of two inequalities joined by 'and' or 'or'.

Graphing Linear Inequalities

  • To graph a system of inequalities, follow these steps:

    • Step 1: Graph the boundary line of each inequality.

    • Use methods such as:

      • Creating a table of values

      • Finding intercepts

      • Applying slope-intercept form

    • Step 2: Determine if the boundary line is solid or dashed.

    • Solid line: Indicates that points on the line are included in the solution (≥ or ≤).

    • Dashed line: Indicates that points on the line are excluded ( > or <).

Example: Graphing x + y ≥ 1

  • The boundary line for the inequality x + y = 1 can be rewritten as:

    • y = -x + 1 (slope-intercept form)

  • Testing Ordered Pairs:

    • Test point (−3, 0):

    • x + y = -3 + 0 = -3 \text{ (FALSE)}

    • Test point (4, 1):

    • x + y = 4 + 1 = 5 \text{ (TRUE)}

  • Shade the region that includes the point (4, 1).

Graphing Second Inequality

Example: Graphing y - x ≥ 5

  • The boundary line is y - x = 5, which can be rewritten as:

    • y = x + 5 (solid line)

  • Testing Points:

    • Test point (−3, 0):

    • y - x = 0 - (-3) = 3 \text{ (FALSE)}

    • Test point (0, 6):

    • y - x = 6 - 0 = 6 \text{ (TRUE)}

  • Shade the region that satisfies the second inequality.

Combined Solution Set

  • The graph of the combination of both inequalities (x + y \text{ and } y - x) shows the area where both shaded regions overlap.

  • The purple shaded region on the graph represents the solution set for the system:

    • Includes all points that satisfy both inequalities.

Compound Inequalities

  • A compound inequality can be treated like two separate inequalities when graphing.

  • Example: Graph the system 3x + 2y < 12 and −1 ≤ y ≤ 5.

    • For the first inequality:

    • The boundary line 3x + 2y = 12 is dashed because it does not include equality.

    • For the second inequality:

    • This translates to two inequalities: y \ge -1 \text{ and } y \le 5.

    • Both boundaries are solid since they include the equalities.

  • Shade the region between the two horizontal lines for the second inequality and the region below the boundary from the first inequality.

  • The overlapping area represents the solution to the entire system.

Summary of Graphing Process

  1. Graph each inequality one at a time, determining the type of boundary line necessary (solid or dashed).

  2. Test points to determine which region of the graph to shade.

  3. Identify the intersection of shaded areas to represent the solution region for the system of inequalities.

  4. For compound inequalities, treat them as separate but ensure to indicate where solutions overlap on the graph.