Session 02-01 - Equations & Inequalities

Section 02: Equations & Problem-Solving Strategies

Author

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz

Quick Review from Section 01

10 minutes - individual work, then we review

  1. Factor completely: \(x^6 - 7x^3 - 8\)

  2. Simplify: \(\frac{(3x^{-2}y^3)^{-2} \cdot (2x^3y^{-1})^3}{6x^{-4}y^2}\)

  3. If \(2^{x+1} + 2^x = 24\), find \(x\)

  4. Rationalize and simplify: \(\frac{3}{\sqrt{5} + \sqrt{3}} - \frac{2}{\sqrt{5} - \sqrt{3}}\)

. . .

Present your solutions and we review together!

Homework Presentations

Solutions Showcase

20 minutes - presentations and discussion

  • Discuss your most challenging problem from Tasks 01-06
  • Share your problem-solving approach
  • Show potential alternative methods
  • Ask questions about problems you found difficult

. . .

Remember: Discussing tasks helps solidify your own understanding!

Key Concept Review

The IDEA Method

A method to help you assess tasks

  • Identify: What type of problem are we solving?
  • Develop: Create a plan using appropriate methods
  • Execute: Carry out the solution carefully
  • Assess: Check your answer makes sense

. . .

Today we apply IDEA to translating word problems into equations and inequalities!

Mathematical Language

Translation Fundamentals

Converting words to mathematical expressions

English Phrase Symbol Example
“is”, “equals”, “is equal to” = “The cost is €50” → \(C = 50\)
“less than”, “fewer than” < “x is less than 10” → \(x < 10\)
“at least”, “no less than” ≥ “at least 5 units” → \(x ≥ 5\)
“at most”, “no more than” ≤ “at most 100” → \(x ≤ 100\)
“increased by”, “plus” + “price increased by €5” → \(p + 5\)
“decreased by”, “minus” - “reduced by 20%” → \(x - 0.2x\)
“of”, “times” × “30% of sales” → \(0.3S\)

Business Vocabulary Essentials

Key terms you’ll encounter frequently

  • Revenue (R): Total income = Price × Quantity
  • Cost (C): Fixed costs + Variable costs
  • Profit (P): Revenue - Cost = R - C
  • Break-even: When Revenue = Cost (Profit = 0)
  • Margin: Profit as percentage of revenue
  • Markup: Increase from cost to selling price

. . .

Always define your variables clearly before translating!

Practice IDEA with Tasks

Lets practice this! Try these on your own

Translate each phrase into an equation and solve:

  1. “Seven more than twice a number equals 31”

  2. “The quotient of a number and 4, decreased by 3, is 12”

  3. “40% of a number increased by 25 equals the number itself”

Break - 10 Minutes

Recap: Solving Multi-Step Equations

A systematic approach

  1. Clear fractions: Multiply by LCD
  2. Expand: Remove parentheses using distributive property
  3. Collect terms: Variables on one side, constants on other
  4. Isolate variable: Divide by coefficient
  5. Verify: Substitute back into original equation

Example: Equation with Fractions

Let’s work through this together

Solve: \(\frac{2x - 1}{3} + \frac{x + 2}{4} = 5\)

  • Step 1: Find LCD → LCD = 12
  • Step 2: Clear fractions → \(12 \cdot \frac{2x - 1}{3} + 12 \cdot \frac{x + 2}{4} = 12 \cdot 5\)
  • Step 3: Simplify → \(4(2x - 1) + 3(x + 2) = 60\)
  • Step 4: Expand → \(8x - 4 + 3x + 6 = 60\)
  • Step 5: Combine → \(11x + 2 = 60\)
  • Step 6: Solve → \(11x = 58\), so \(x = \frac{58}{11}\)

Recap: Inequalities

When things aren’t necessaryly equal

  • When multiplying or dividing by negative number, flip the sign!
    • Example: \(-2x > 6\)
    • Divide by -2: \(x < -3\) (sign flipped!)
    • Why? Because the number line reverses!
  • Inequalities are used to restrict the range of a variable
  • Often Used to bound the solution space in business applications

Example: Business Application

Profit constraints in action

A company has costs \(C = 5000 + 20x\) and revenue \(R = 50x\).

How many units must they sell to make at least €4000 profit?

  • Set up: Profit = Revenue - Cost ≥ 4000
  • Equation: \(50x - (5000 + 20x) ≥ 4000\)
  • Simplify: \(30x - 5000 ≥ 4000\)
  • Solve: \(30x ≥ 9000\), so \(x ≥ 300\)
  • Answer: Must sell at least 300 units

Practice

Individual Exercises

Work independently, then we’ll discuss

  1. To equation: “Three times a number decreased by 7 equals 14”

  2. Solve: \(3(2x - 4) = 2(x + 5)\)

  3. Solve the inequality: \(-3x + 7 < 16\)

  4. A taxi charges €3.50 base fare plus €1.20 per km. If a ride costs €15.50, how far was it?

  5. A store offers 30% discount. After discount, an item costs €42. What was the original price?

Application & Extension

Break-Even Analysis

Where total revenue equals total cost (profit = 0)

A coffee shop has fixed costs of €2,000/month (rent, utilities), variable cost of €1.50 per coffee and a selling price of €3.50 per coffee. How many coffees for break-even?

  • Let \(x\) = number of coffees
  • Cost: \(C = 2000 + 1.50x\)
  • Revenue: \(R = 3.50x\)
  • Break-even: \(3.50x = 2000 + 1.50x\)
  • Solve: \(2x = 2000\), so \(x = 1000\) coffees

Mixture Problems

Combining different concentrations or values

An investor has €10,000 to split between bonds (4% return) and stocks (9% return). To earn €650 annually, how much in each?

  • Let \(x\) = amount in bonds
  • Then \(10000 - x\) = amount in stocks
  • Income equation: \(0.04x + 0.09(10000 - x) = 650\)
  • Simplify: \(0.04x + 900 - 0.09x = 650\)
  • Solve: \(-0.05x = -250\), so \(x = 5000\)
  • Answer: €5,000 in bonds, €5,000 in stocks

Coffee Break - 15 Minutes

Collaborative Problem-Solving

Group Task

Work in groups on the following problem

A company produces two products:

  • Product A: Costs €15 to make, sells for €25
  • Product B: Costs €20 to make, sells for €35
  • Fixed costs: €5,000/month
  • Production capacity: 500 units total
  • Must produce at least 100 of each product

The tasks

Work in groups on the following problem

  1. Set up the profit equation
  2. Find the break-even point if producing equal quantities
  3. What mix maximizes profit?

Wrap-up & Synthesis

Key Takeaways

Essential skills from today

  • Translation from words to equations is systematic
  • Multi-step equations require organized approach
  • Inequalities have special rules (flip when multiplying by negative!)
  • Business problems often involve setting up profit/cost equations
  • Break-even analysis is fundamental to business planning

Common Pitfalls to Avoid

Watch out for these!

  • Forgetting to flip inequality signs
  • Misinterpreting “less than” in word problems
  • Not checking solutions in original equation
  • Mixing up revenue and profit
  • Forgetting units in final answers

Final Assessment

Individual work

A small business has monthly costs of €3,000 plus €12 per unit produced. They sell each unit for €20.

  1. Write the profit equation
  2. How many units for break-even?
  3. How many units for €2,000 profit?

Next Session Preview

Session 02-02: Systems of Equations

  • Solving systems by substitution and elimination
  • Business applications with multiple constraints
  • Introduction to linear programming

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TipPreparation Tip

Review today’s equation-solving techniques - they’re the foundation for systems!