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Reference Guide 6.2: Graphing Systems of Linear Inequalities

Key Terms

• Systems of Linear Inequalities: A set of two or more linear inequalities graphed on a coordinate plane; the intersection of their solution regions represents the solution set.

Important Information

Steps for Solving Systems of Inequalities by Graphing

1. Formulate Inequalities: Convert the word problem into two applicable inequalities.

2. Graph Each Inequality: Use preferred graphing methods (see Reference Guide 1).

3. Identify Overlapping Region: Highlight the region where both inequalities overlap.

4. Select Potential Solutions: All points within the overlapping region are potential solutions; choose appropriate answers.

Practice Problems

1. Graphing Exercise: Graph the system:

   • Inequality 1: ( y > 2x - 1 )

   • Inequality 2: ( -y < 2 )

   • Graph: Shade the shared region and find three possible solutions.

   • Graphing Area: x-values range from -10 to 10; y-values from -10 to 10.

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2. Graphing Exercise: Graph the system:

   • Inequality 1: ( y - 4 > -2x )

   • Inequality 2: ( -8x + 4y > 12 )

   • Graph: Shade the shared region.

3. Graphing Exercise: Graph the system:

   • Inequality 1: ( 2x + 3y \geq 9, x \in , real, y \in , real )

   • Inequality 2: ( y - 6x \leq 1, x \in , real, y \in , real )

   • Graph: Shade the shared region and implement three possible solutions.

Practice Questions

• Worksheet on Solving Systems of Linear Inequalities.

Reference Guide 6.1: Graphing Linear Inequalities

Key Terms

• x-intercept: The point where a line crosses the x-axis (y=0).

• y-intercept: The point where a line crosses the y-axis (x=0).

Important Information

Graphing Techniques

• Using Intercepts:

  • Find x and y intercepts by substituting 0 for y (to find x-intercept) and for x (to find y-intercept).

  • Connect points with a ruler to form a line.

• Using Slope-Intercept Form:

  • Rearrange the equation to isolate y.

  • Plot the y-intercept and use the slope to determine additional points.

Graphing Symbols

• Comparison Symbols:

  • < : less than

  • ≤ : less than or equal to

  • : greater than

  • ≥ : greater than or equal to

• Graphing Lines:

  • Use a dotted line for < or >; solid line for ≤ or ≥.

• Shading Areas:

  • Shade below line for < and ≤; above line for > and ≥.

Considerations for Counting Numbers

• Use stippled boundaries instead of lines for counting numbers.

Page 4: Practice Problems

1. Graph the Solution Set: ( -2x + 5y \leq 10 )

2. Convert to Slope-Intercept Form: Find and graph the inequality representation.

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3. Word Problem Representation:

   • A sports store has a net revenue of $100 on each pair of downhill skis and $120 on snowboards.

   • Formulate inequality based on daily sales goal: ( 100x + 120y > 600 ).

   • Plot graph showing intercepts, state two combinations of ski and snowboard sales exceeding this goal.

Practice Questions

• Refer to page 303: problems #1-6,8,9,10.

Reference Guide 6.3: Graphing to Solve Systems of Linear Inequalities

Key Terms

• No new terms in this section.

Important Information

Steps for Solving Systems of Inequalities by Graphing

1. Formulate Two Inequalities based on the problem statement.

2. Graph Each Inequality following techniques in Reference Guide 1.

3. Identify Overlapping Region for potential solutions.

Practice Problems

1. Boat Production Problem:

   • Inequalities:

     • ( x + y \leq 20 ) (Total boats)

     • ( y ≥ x + 5 ) (More fiberglass boats)

   • Graph and choose points from the overlap.

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2. Graph the Solution Set: Inequalities specified: ( 3x + 2y > -y )

3. Sailing Problem:

   • Describes limits on mainsail and jib combinations based on sailing conditions.

Practice Questions

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• Refer to page 317: problems (#1,2,4,6,7,8,9).

Reference Guide 6.4: Optimization Problems

Key Terms

• Optimization Problem: Maximizing or minimizing a quantity based on constraints.

• Constraint: Conditions limiting the optimization problem.

• Feasible Region: Area representing solutions to inequalities modeling the problem.

Important Information

Steps for Solving Optimization Problems

1. List all Constraints.

2. Graph Linear Inequalities based on constraints.

3. Highlight Feasible Region produced by inequalities.

4. Identify Corner Coordinates of feasible region as potential solutions.

5. Create Optimization Equation.

6. Evaluate Each Corner Point in the optimization equation; identify max/min values.

Practice Problems

1. Toy Company Production:

   • Constraints on numbers of racing cars and SUVs.

   • Identify combinations for minimal and maximal production costs.

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2. Maximization Problem: L & G Construction Scenario.

   • Fence constraints regarding wood types and costs outlined; find costs for construction.

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3. Refinery Production Model:

   • Determine daily production mix for maximizing revenue with constraints on gasoline & heating oil.

Practice Questions

• Refer to Optimization Problems Worksheet, Page 341 (#1,3,5,6,11,12,13,15).