Linear inequalities in two variables and linear programming

Introduction to Linear Inequalities and Programming

Welcome to one of the most practical chapters in Further Pure Mathematics! Linear Programming is a powerful tool used by businesses and engineers to make the best possible decisions—like how to get the most profit or use the least amount of material. At its heart, this topic is about drawing lines on a graph and finding the "sweet spot" where all your conditions are met.

Don't worry if you find graphs a bit intimidating; we will break this down into simple, manageable steps. By the end of these notes, you will be able to turn word problems into shaded regions and find the perfect solution every time.

1. Understanding Linear Inequalities

Before we can program, we need to know how to describe boundaries. A linear inequality looks just like a straight-line equation, but instead of an equals sign \( (=) \), it uses one of these four symbols:

• \( < \) : Less than (indicated by a dashed line)
• \( > \) : Greater than (indicated by a dashed line)
• \( \leq \) : Less than or equal to (indicated by a solid line)
• \( \geq \) : Greater than or equal to (indicated by a solid line)

The "Boundary Line" Rule

When graphing an inequality like \( y \leq 2x + 3 \), your first step is always to imagine it as the line \( y = 2x + 3 \). This line acts as a "fence" separating the coordinate plane into two sides.

Quick Review: Remember the forms of a straight line from Section 8 of your syllabus:
\( y = mx + c \) (Gradient-intercept form)
\( ax + by = c \) (General form)

2. How to Graph an Inequality

Follow these three simple steps to graph any linear inequality:

Step 1: Draw the boundary line.
Find two or three points that satisfy the equation. For example, for \( x + y = 4 \), you could use \((0, 4)\) and \((4, 0)\). Connect them with a straight edge.

Step 2: Choose your line type.
- Use a solid line if the symbol is \( \leq \) or \( \geq \). This means points on the line are included.
- Use a dashed line if the symbol is \( < \) or \( > \). This means points on the line are not included.

Step 3: Shade the correct region.
Pick a "test point" that is not on the line—the origin \((0, 0)\) is usually the easiest choice! Plug the \( x \) and \( y \) values into your inequality.
- If the inequality is true (e.g., \( 0 < 5 \)), shade the side containing \((0, 0)\).
- If the inequality is false (e.g., \( 0 > 5 \)), shade the opposite side.

Note for Exam Success: Read the question carefully! Some examiners ask you to shade the wanted region, while others ask you to shade the unwanted region to leave the "Feasible Region" clear. Always check the instructions.

3. Defining the Feasible Region

In most exam questions, you will be given a set of several inequalities (called constraints). The Feasible Region (often labeled \( R \)) is the area on the graph where all the inequalities are true at the same time.

Common Constraints to Watch For:
- \( x \geq 0 \) and \( y \geq 0 \): This simply means your region must be in the top-right quadrant (where everything is positive).
- \( x = k \): A vertical line.
- \( y = k \): A horizontal line.

Key Takeaway: The Feasible Region is the overlap of all your shaded areas. Any point inside this region is a "possible" solution to the problem.

4. Introduction to Linear Programming

Linear Programming is the process of finding the maximum or minimum value of an Objective Function within the Feasible Region. An objective function usually looks like this:
\( P = ax + by \)

For example, if \( x \) is the number of apples sold and \( y \) is the number of bananas, and you make \$2 profit per apple and \$3 per banana, your objective function is:
\( P = 2x + 3y \)

Finding the Optimal Point

The "Best" solution (the maximum or minimum) will almost always occur at one of the vertices (corners) of the Feasible Region. Here is how to find it:

The Corner Point Method:
1. Identify the coordinates of all corner points of your region \( R \).
2. If a corner is the intersection of two lines, solve them as simultaneous equations to get the exact \( (x, y) \).
3. Plug each coordinate into your objective function \( P = ax + by \).
4. Compare the results. The highest value is your maximum; the lowest is your minimum.

The Objective Line (Search Line) Method:
1. Set your objective function to a random constant, e.g., \( ax + by = 10 \).
2. Draw this line on your graph (usually with a dotted line).
3. Use a ruler to move this line parallel across the Feasible Region.
4. The last point the line touches as it leaves the region is the maximum. The first point it touches is the minimum.

5. Common Pitfalls and Tips

• Integer Constraints: If the question is about people, cars, or cakes, your answer must be a whole number. If the vertex is \( (2.4, 3.8) \), you must check the nearest whole-number points inside the region.
• Incorrect Shading: Double-check your test point! If you test \( (0, 0) \) and get \( 0 \leq 10 \), that is true, so \((0,0)\) is in the "good" zone.
• Simultaneous Equations: Don't just guess the coordinates from the graph. Solve the equations of the two lines (e.g., using elimination or substitution) to be 100% accurate.

Summary Checklist

• Boundary Lines: Are they solid (\( \leq, \geq \)) or dashed (\( <, > \))?
• Feasible Region: Is the region \( R \) clearly identified where all constraints overlap?
• Objective Function: Have you correctly identified the expression you are trying to maximize or minimize?
• Verification: Have you checked the vertices of the region to find the optimal value?

Don't worry if this seems tricky at first! Linear programming is just a visual way of solving algebra. Once you get used to the "test point" method, you'll find it's very logical.