Linear Inequalities and Applications

TL;DR

Linear inequalities show relationships where one expression is greater than, less than, or equal to another. You'll solve them like equations but flip the inequality sign when multiplying or dividing by negative numbers. They're essential for modeling real-world constraints like budgets, time limits, and resource allocation.

1. The Mental Model

Think of a linear inequality as drawing a line on a graph, then shading one side to show all the solutions that work. Unlike equations that give you one answer, inequalities give you a whole region of possibilities. That's the whole idea.

2. The Core Material

Basic Inequality Solving

You solve linear inequalities almost exactly like linear equations. Add, subtract, multiply, and divide to isolate your variable. There's just one crucial rule: when you multiply or divide both sides by a negative number, flip the inequality sign.

Let's see why this matters:
- Start with: 5 > 3 (true)
- Multiply both sides by -1: -5 > -3 (false!)
- Flip the sign: -5 < -3 (true again)

Here's your step-by-step process:
1. Simplify both sides (distribute, combine like terms)
2. Move variables to one side, constants to the other
3. Divide by the coefficient of your variable
4. If that coefficient is negative, flip the inequality sign

Example: Solve -3x + 7 ≥ 16
- Subtract 7: -3x ≥ 9
- Divide by -3 (flip the sign): x ≤ -3

Graphing Solutions

When you solve x ≤ -3, you're not done yet. You need to understand what this means visually. On a number line, you'd put a solid dot at -3 (because x can equal -3) and shade everything to the left.

The inequality symbols tell you exactly what to draw:
- < or >: Open circle (value not included)
- ≤ or ≥: Solid dot (value included)
- <: Shade left
- >: Shade right

For two-variable inequalities like y < 2x + 1, you'll graph the boundary line y = 2x + 1, then shade the region that satisfies the inequality. Test a point (usually the origin if it's not on the line) to see which side to shade.

Real-World Applications

This is where inequalities become powerful. Most real problems don't have exact answers – they have constraints and ranges of acceptable solutions.

Budget problems: "You have $50 to spend on books costing $12 each and notebooks costing $3 each." If x = books and y = notebooks, then 12x + 3y ≤ 50.

Time constraints: "A project takes 2 hours per task A and 3 hours per task B, with at most 40 hours available." That's 2A + 3B ≤ 40.

Production limits: "A factory can make at most 500 widgets per day, with morning shift producing twice as many as afternoon shift." If morning = M and afternoon = A, then M + A ≤ 500 and M = 2A.

The beauty is that these problems often have multiple correct answers. Your job is finding the feasible region – all the combinations that satisfy every constraint simultaneously.

3. Worked Example

Let's solve a real budget problem from start to finish.

Problem: Sarah has $100 to buy concert tickets. Premium tickets cost $25 each, regular tickets cost $15 each. She wants at least 2 premium tickets. How many of each type can she buy?

Step 1: Define variables and constraints
- Let p = premium tickets, r = regular tickets
- Budget constraint: 25p + 15r ≤ 100
- Minimum premium: p ≥ 2
- Non-negative: p ≥ 0, r ≥ 0

Step 2: Find the boundary line for the budget
25p + 15r = 100
Solve for r: r = (100 - 25p)/15 = 20/3 - 5p/3

Step 3: Test feasible combinations
With p = 2: 25(2) + 15r ≤ 100 → 50 + 15r ≤ 100 → r ≤ 10/3 ≈ 3.33
So r can be 0, 1, 2, or 3.

With p = 3: 25(3) + 15r ≤ 100 → 75 + 15r ≤ 100 → r ≤ 25/15 ≈ 1.67
So r can be 0 or 1.

With p = 4: 25(4) = 100, so r = 0 only.

Answer: Valid combinations are (2,0), (2,1), (2,2), (2,3), (3,0), (3,1), and (4,0).

4. Key Takeaways

4.1 Most Important Concepts

• Sign flipping rule: Always flip the inequality when multiplying/dividing by negative numbers
• Boundary behavior: Use solid dots/lines for ≤ and ≥, open circles/dashed lines for < and >
• Solution sets: Inequalities typically have infinitely many solutions forming regions or intervals
• Feasible regions: Real problems often combine multiple inequalities to define acceptable solution areas
• Testing points: Always verify your shaded region by testing a point that should satisfy the inequality
• Integer constraints: Many real problems require whole number solutions within the feasible region

4.2 Common Misconceptions

• "I can ignore the sign flip" → This creates completely wrong solution sets; the rule is non-negotiable
• "The solution is just one number" → Unlike equations, inequalities give you ranges or regions of solutions
• "All points in my shaded region work" → Real problems often add integer or non-negative constraints
• "I can solve compound inequalities by splitting them" → You must solve the entire system simultaneously to find the intersection

4.3 Compare & Contrast

5. Now Try It

Solve this planning problem: "A student has 20 hours to study for two exams. Math needs at least 8 hours, English needs at least 5 hours. Math is twice as important, so math hours should be at least 1.5 times English hours. Find all valid time allocations."

Set up your variables, write the constraint inequalities, and find 3 specific combinations that work. Success looks like: correct inequalities, proper solving steps, and three realistic hour combinations that satisfy all constraints.