Steps to Write Inequalities from Word Problems:

1. Step 1: Identify the variables
   • Determine what x and y represent
   • Define them clearly
2. Step 2: Identify the constraint words
   • "At most", "no more than", "maximum" → \( \leq \)
   • "At least", "no less than", "minimum" → \( \geq \)
   • "Less than", "fewer than" → \( < \)
   • "More than", "greater than" → \( > \)
3. Step 3: Write the expression
   • Translate the relationship into mathematical form
4. Step 4: Write the inequality
   • Combine the expression with the appropriate inequality symbol

Common Key Words & Symbols:

Example 1: Budget Problem

"Sarah wants to buy apples at $3 per pound and bananas at $2 per pound. She has at most $24 to spend. Write an inequality."

Step 1: Let x = pounds of apples, y = pounds of bananas

Step 2: "At most $24" means \( \leq 24 \)

Step 3: Cost = 3x + 2y

Answer: \( 3x + 2y \leq 24 \)

Example 2: Production Problem

"A factory produces chairs and tables. Each chair takes 2 hours and each table takes 4 hours. The factory has more than 40 hours available. Write an inequality."

Step 1: Let x = number of chairs, y = number of tables

Step 2: "More than 40 hours" means \( > 40 \)

Step 3: Time = 2x + 4y

Answer: \( 2x + 4y > 40 \)

Example 3: Capacity Problem

"A truck can carry at least 50 boxes of type A (weighing 10 kg each) and type B (weighing 15 kg each). Write an inequality for the number of boxes."

Step 1: Let x = boxes of type A, y = boxes of type B

Step 2: "At least 50 boxes" means \( \geq 50 \)

Answer: \( x + y \geq 50 \)

Steps to Graph a Two-Variable Inequality:

1. Step 1: Graph the boundary line
   • Replace the inequality symbol with = to get the equation
   • Graph this line using slope-intercept or any method
   • Use dashed line for < or > (boundary NOT included)
   • Use solid line for ≤ or ≥ (boundary IS included)
2. Step 2: Choose a test point
   • Use (0, 0) if it's not on the boundary line
   • If (0, 0) is on the line, use (0, 1) or (1, 0)
3. Step 3: Test the point
   • Substitute the test point into the original inequality
   • Check if it makes the inequality true
4. Step 4: Shade the appropriate region
   • If test point makes inequality TRUE: shade the side containing the test point
   • If test point makes inequality FALSE: shade the opposite side

Quick Shading Guide (Slope-Intercept Form):

When inequality is in form \( y < mx + b \) or \( y > mx + b \):

• \( y > mx + b \) or \( y \geq mx + b \): Shade ABOVE the line
• \( y < mx + b \) or \( y \leq mx + b \): Shade BELOW the line

Example 1: Graph \( y \leq 2x + 1 \)

Step 1: Graph the boundary line \( y = 2x + 1 \) using a solid line (because of ≤)

• y-intercept: 1 (point: 0, 1)

• Slope: 2 (rise 2, run 1)

Step 2: Test point (0, 0)

Step 3: \( 0 \leq 2(0) + 1 \) → \( 0 \leq 1 \) ✓ TRUE

Step 4: Shade the region containing (0, 0), which is below the line

Example 2: Graph \( x + 2y > 6 \)

Step 1: Rewrite: \( y > -\frac{1}{2}x + 3 \)

Step 2: Graph \( y = -\frac{1}{2}x + 3 \) using a dashed line (because of >)

Step 3: Test (0, 0): \( 0 + 2(0) > 6 \) → \( 0 > 6 \) ✗ FALSE

Step 4: Shade the opposite side (away from origin), which is above the line

Steps to Check if a Point is a Solution:

1. Step 1: Substitute the x and y values into the FIRST inequality
2. Step 2: Check if the first inequality is true
3. Step 3: Substitute the same values into the SECOND inequality
4. Step 4: Check if the second inequality is true
5. Step 5: Repeat for ALL inequalities in the system
6. Result: The point is a solution ONLY if it satisfies ALL inequalities simultaneously

Important Notes:

• A point must satisfy ALL inequalities to be a solution
• If it fails even one inequality, it is NOT a solution
• The solution set is the intersection of all individual solution regions

Example 1: Is (2, 3) a solution?

System: \( \begin{cases} y \leq 2x + 1 \\ y > x \end{cases} \)

Check Inequality 1: \( 3 \leq 2(2) + 1 \) → \( 3 \leq 5 \) ✓ True

Check Inequality 2: \( 3 > 2 \) ✓ True

Answer: YES, (2, 3) is a solution (satisfies both inequalities)

Example 2: Is (1, 4) a solution?

System: \( \begin{cases} x + y < 6 \\ 2x - y \geq 0 \end{cases} \)

Check Inequality 1: \( 1 + 4 < 6 \) → \( 5 < 6 \) ✓ True

Check Inequality 2: \( 2(1) - 4 \geq 0 \) → \( -2 \geq 0 \) ✗ False

Answer: NO, (1, 4) is NOT a solution (fails second inequality)

Steps to Solve a System by Graphing:

1. Step 1: Graph the first inequality
   • Draw the boundary line (solid or dashed)
   • Shade the appropriate region
2. Step 2: Graph the second inequality on the same axes
   • Draw the boundary line
   • Shade the appropriate region
3. Step 3: Identify the overlapping region
   • The solution is where the shaded regions OVERLAP
   • This is called the feasible region
4. Step 4: Verify with a test point
   • Pick a point from the overlapping region
   • Check it satisfies all inequalities

Key Terminology:

• Feasible Region: The overlapping shaded area (solution set)
• Boundary Lines: Lines that define the edges of the region
• Vertices: Corner points where boundary lines intersect
• Unbounded Region: Region that extends infinitely in one or more directions

Example: Solve the System

System: \( \begin{cases} y \leq x + 2 \\ y \geq -x + 1 \end{cases} \)

Step 1: Graph \( y = x + 2 \) with solid line, shade below

Step 2: Graph \( y = -x + 1 \) with solid line, shade above

Step 3: Identify overlap (region between the two lines)

Solution: The region where both shadings overlap, bounded by both lines

Special Cases:

No Solution: If the shaded regions don't overlap at all (parallel lines with opposite shading)

Infinite Solutions: If inequalities describe the same region

Bounded Region: If the feasible region is enclosed (like a polygon)

What are Vertices?

Vertices (or corner points) are the points where two or more boundary lines intersect. They form the corners of the feasible region.

Vertices are critical in linear programming because optimal values often occur at these points.

Steps to Find Vertices:

1. Step 1: Graph the system of inequalities
   • Draw all boundary lines
   • Identify the feasible region
2. Step 2: Identify intersection points
   • Find where boundary lines intersect
   • Mark all corner points of the feasible region
3. Step 3: Solve for each vertex
   • Take pairs of boundary line equations
   • Solve the system to find exact coordinates
   • Use substitution or elimination method
4. Step 4: Verify vertices are in feasible region
   • Check each vertex satisfies ALL inequalities
   • Discard any that don't satisfy all constraints

Example: Find All Vertices

System: \( \begin{cases} x \geq 0 \\ y \geq 0 \\ x + y \leq 6 \\ 2x + y \leq 10 \end{cases} \)

Boundary lines:

• \( x = 0 \) (y-axis)

• \( y = 0 \) (x-axis)

• \( x + y = 6 \)

• \( 2x + y = 10 \)

Find vertices (intersection points):

Vertex 1: \( x = 0 \) and \( y = 0 \) → (0, 0)

Vertex 2: \( x = 0 \) and \( x + y = 6 \) → \( y = 6 \) → (0, 6)

Vertex 3: \( y = 0 \) and \( 2x + y = 10 \) → \( 2x = 10 \) → (5, 0)

Vertex 4: Solve \( \begin{cases} x + y = 6 \\ 2x + y = 10 \end{cases} \)

Subtract: \( x = 4 \), then \( y = 2 \) → (4, 2)

Vertices: (0, 0), (0, 6), (5, 0), (4, 2)

⚠️ Important Notes:

• Not all intersection points are vertices - only those in the feasible region
• Always verify each vertex satisfies all inequalities
• The feasible region can have 3, 4, 5, or more vertices depending on the system

What is Linear Programming?

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

Real-world applications: Profit maximization, cost minimization, resource allocation, production planning

Key Components:

Steps to Solve a Linear Programming Problem:

1. Step 1: Define variables
   • Let x and y represent the unknowns
2. Step 2: Write the objective function
   • Write the function to maximize or minimize
   • Usually in form \( P = ax + by \) or \( C = ax + by \)
3. Step 3: Write the constraints
   • Write all inequalities from the problem
   • Include non-negativity constraints (\( x \geq 0, y \geq 0 \))
4. Step 4: Graph the feasible region
   • Graph all constraints
   • Identify the feasible region
5. Step 5: Find all vertices
   • Find coordinates of all corner points
6. Step 6: Evaluate objective function at each vertex
   • Substitute each vertex into the objective function
   • Calculate the value at each corner point
7. Step 7: Identify optimal solution
   • Maximum: Choose vertex with largest value
   • Minimum: Choose vertex with smallest value

Fundamental Theorem of Linear Programming:

If an optimal solution exists, it will occur at one or more vertices of the feasible region.

Example: Profit Maximization

"A company makes two products: A and B. Product A gives $5 profit and B gives $7 profit. Product A requires 2 hours and B requires 3 hours. Maximum 120 hours available. Also, at most 40 units of A and 30 units of B can be made. Find maximum profit."

Step 1: Let x = units of A, y = units of B

Step 2: Objective function: Maximize \( P = 5x + 7y \)

Step 3: Constraints:

• \( 2x + 3y \leq 120 \) (time)

• \( x \leq 40 \) (limit on A)

• \( y \leq 30 \) (limit on B)

• \( x \geq 0, y \geq 0 \) (non-negative)

Step 4-5: Graph and find vertices: (0, 0), (40, 0), (0, 30), (40, 13.33), (30, 30)

Step 6: Evaluate P at each vertex:

• (0, 0): P = 0

• (40, 0): P = 200

• (0, 30): P = 210

• (40, 13.33): P = 293.31

• (30, 30): P = 360 (Maximum!)

Answer: Make 30 units of A and 30 units of B for maximum profit of $360

Common Applications:

• Manufacturing: Maximize profit while minimizing costs
• Agriculture: Optimize crop planting for maximum yield
• Transportation: Minimize shipping costs
• Nutrition: Minimize cost while meeting nutritional requirements
• Finance: Maximize return on investment with budget constraints

Key Concepts:

• Write Inequalities: Identify variables, translate key words to symbols (at most = ≤, at least = ≥)
• Graph Inequalities: Draw boundary line (solid/dashed), test point, shade correct region
• Check Solutions: Point must satisfy ALL inequalities in the system
• Systems: Solution is the overlapping (feasible) region
• Vertices: Corner points where boundary lines intersect in feasible region
• Linear Programming: Evaluate objective function at all vertices to find optimal solution