The solution to a system of inequalities is the region where all shaded areas overlap

1. Graph the first inequality:

a) Replace inequality symbol with = to get boundary line

b) Determine if line is solid or dashed

c) Test a point (usually (0,0) if not on line) to determine shading

2. Graph the second inequality:

Repeat the same process on the same coordinate plane

3. Identify the solution region:

The overlapping shaded area is the solution

4. Verify:

Pick a point in the solution region and test in all inequalities

Solve by graphing:

\[ \begin{cases} y \leq 2x + 1 \\ y > -x + 3 \end{cases} \]

Two Cases:

1. Graph the boundary: Replace inequality with equality \( y = |ax + b| + c \)

2. This creates a V-shaped graph (vertex at the turning point)

3. Determine solid or dashed boundary

4. Shade inside V for \( \leq \) or \( < \); outside V for \( \geq \) or \( > \)

Solve by graphing:

\[ \begin{cases} y \leq |x - 2| + 1 \\ y > x - 3 \end{cases} \]

Vertices (corner points) are the intersection points that form the corners of the feasible region (solution set)

1. Graph the system of inequalities to identify the feasible region

2. Identify intersection points of boundary lines that form corners

3. Solve systems of equations for each pair of intersecting boundaries

4. Test each point to verify it satisfies all inequalities

5. List all vertices as ordered pairs

Find vertices of the feasible region:

\[ \begin{cases} x + y \leq 6 \\ 2x + y \leq 8 \\ x \geq 0 \\ y \geq 0 \end{cases} \]

Linear programming is a method to find the maximum or minimum value of a linear function subject to constraints (system of linear inequalities)

If a maximum or minimum value exists, it occurs at one or more vertices of the feasible region

1. Define variables and write the objective function

2. Write constraints as a system of inequalities

3. Graph the feasible region and find vertices

4. Evaluate the objective function at each vertex

5. Identify the optimal solution (largest for max, smallest for min)

Maximize: \( P = 3x + 4y \)

Subject to constraints:

\[ \begin{cases} x + 2y \leq 10 \\ x + y \leq 6 \\ x \geq 0 \\ y \geq 0 \end{cases} \]

• Maximizing profit given production constraints

• Minimizing cost with resource limitations

• Optimizing nutrition with dietary requirements

• Resource allocation problems

• Transportation and scheduling optimization

Bounded Region:

• Enclosed polygon with finite vertices

• Both maximum and minimum values exist

Unbounded Region:

• Extends infinitely in one or more directions

• May have no maximum or no minimum

Empty Set:

• No region of overlap (inconsistent system)

• No solution exists

• Always test points to verify shading direction

• Use different colors/patterns for each inequality when possible

• Check all vertices systematically in linear programming

• Be careful with dashed vs solid boundaries

• Verify vertices satisfy all constraints before evaluating objective function