Session 01-06 - Synthesis & Linear Equations

Section 01: Mathematical Foundations & Algebra

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz

Quick Skills Check

Test your readiness

  1. Simplify: \((2x^3)^2 \cdot x^{-5}\)

  2. Factor: \(x^2 - 5x - 14\)

  3. Evaluate: \(\log_2(32)\)

  4. Simplify: \(\sqrt{48}\)

  5. Express in scientific notation: \(0.000425\)

These cover the core skills - let’s review together!

Homework Discussion

Showcase Your Solutions

Comprehensive discussions

  • Present your most challenging problem from Tasks 01-05
  • Are there any tasks from previous lectures you want to discuss?
  • Explain your problem-solving strategy
  • Share any connections between topics you discovered

Today we see how all pieces fit together!

Learning Objectives

Learning Objectives

By the end of this session, you can:

  • Combine all Section 1 techniques: exponents, factorization, radicals, logarithms
  • Translate word problems into equations using the IDEA method
  • Solve multi-step linear equations and inequalities systematically
  • Apply break-even and mixture analysis to business problems

Key Concepts

Set Theory & Number Systems

Essential foundations

Number hierarchy: \(\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}\)

Set operations:

  • Union: \(A \cup B\) (all elements in either)
  • Intersection: \(A \cap B\) (elements in both)
  • Difference: \(A \setminus B\) (in A but not B)

Quick practice:

  • If \(A = \{1, 3, 5\}\) and \(B = \{3, 4, 5, 6\}\)
  • Find \(A \cap B\) and \(A \cup B\)

Exponent Laws

Your main tools for exponents

Rule Formula Quick Example
Product \(a^m \cdot a^n = a^{m+n}\) \(x^3 \cdot x^4 = x^7\)
Quotient \(\frac{a^m}{a^n} = a^{m-n}\) \(\frac{x^5}{x^2} = x^3\)
Power \((a^m)^n = a^{mn}\) \((x^3)^2 = x^6\)
Negative \(a^{-n} = \frac{1}{a^n}\) \(x^{-2} = \frac{1}{x^2}\)

Quick practice: Simplify \(\frac{(3x^2)^3}{9x^4}\)

Factorization Techniques

  1. Common factor: Always check first!
  2. Difference of squares: \(a^2 - b^2 = (a+b)(a-b)\)
  3. Perfect squares: \(a^2 \pm 2ab + b^2 = (a \pm b)^2\)
  4. AC method: For \(ax^2 + bx + c\) when \(a \neq 1\)
  5. Grouping: For 4-term polynomials
  6. Cubes: \(a^3 \pm b^3 = (a \pm b)(a^2 \mp ab + b^2)\)
  7. Substitution: Let $u = $ common expression

Quick practice: Factor \(x^4 - 16\)

Radicals & Logarithms

Advanced operations

Radicals: extract perfect powers; rationalize with conjugates: \((a + \sqrt{b})(a - \sqrt{b}) = a^2 - b\)

Logarithms:

  • \(\log_a(xy) = \log_a(x) + \log_a(y)\), \(\quad\log_a(x^n) = n\log_a(x)\)
  • If \(a^x = b\), then \(x = \log_a(b)\)

Quick practice: Solve \(2^x = 10\)

Practice Block 1

Individual Exercise

10 minutes - repeat the basics

  1. Simplify: \(\frac{x^3 \cdot x^{-5}}{x^{-3}}\)

  2. Factor completely: \(3x^2 - 27\)

  3. Rationalize: \(\frac{6}{\sqrt{3}}\)

  4. Solve: \(|3x - 6| = 9\)

  5. Evaluate: \(\log_3(81) - \log_3(3)\)

  6. Express in scientific notation: 45,600,000

Individual Exercise - A Step Further

10 minutes, same tools, complicated

  1. Simplify: \(\left(\frac{8x^6}{27}\right)^{2/3}\)

  2. Factor completely: \(6x^2 + x - 12\)

  3. Compute and express in scientific notation: \(\frac{(3.2 \times 10^5)(5 \times 10^{-2})}{8 \times 10^3}\)

  4. Solve: \(\log_5(x + 1) - \log_5(x - 1) = 1\)

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

  6. Simplify for \(x > 0\): \(\sqrt{50x^3}\)

Break

Practice Block 2

Pair Exercise

15 minutes - work together

  1. Factor using substitution: \(x^4 - 5x^2 + 4\)

  2. Simplify: \(\sqrt{75} + \sqrt{48} - \sqrt{27}\)

  3. If \(2^x = 3\) and \(3^y = 5\), find \(2^{xy}\)

  4. Rationalize: \(\frac{4}{3 - \sqrt{5}}\)

  5. Factor by grouping: \(x^3 + 2x^2 - 9x - 18\)

Mixed Technique Problems

Combine multiple methods

  1. Factor: \(8x^3 - 27y^3\)

  2. Simplify: \(\frac{x^3 - 8}{x^2 - 4} \cdot \frac{x + 2}{x^2 + 2x + 4}\)

  3. If \(|2x - 5| < 7\), find the solution interval

  4. Rationalize: \(\frac{2}{\sqrt{6} - \sqrt{2}}\)

Algebra to Equations

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

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

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” → \(x ≤ 100\)

Translation Fundamentals II

English Phrase Symbol Example
“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

  • 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
  • QU, MU: Quantity and monetary units, e.g. 1 QU = 100 pieces

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”

Linear Equations

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 \(x = \frac{58}{11}\)

Inequalities - The Traps

You have solved these since Session 01-03 - now the pitfalls

  • Flip on negative: \(-2x > 6\) gives \(x < -3\), not \(x > -3\)
  • “And” vs “or”: \(|A| < c\) joins with and (one interval), \(|A| > c\) with or (two rays)
  • Never multiply or divide by a variable of unknown sign: from \(\frac{6}{x} > 2\) you cannot conclude \(x < 3\) - for \(x = -1\) the left side is \(-6\)
  • Strict vs inclusive: “at least 300 units” is \(x \geq 300\), but “more than 300” is \(x > 300\)

In business the answer is usually a range of acceptable decisions, not a single number.

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 EUR 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

Coffee Break

Guided 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 EUR base fare plus 1.20 EUR per km. If a ride costs 15.50 EUR, how far was it?

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

Business Applications

Break-Even Analysis

Where total revenue equals total cost (profit = 0)

A coffee shop has fixed costs of 2,000 EUR/month (rent, utilities), variable cost of 1.50 EUR per coffee and a selling price of 3.50 EUR 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

Break-Even, Seen Graphically

Cost line C = 2000 + 1.50x and revenue line R = 3.50x for 0 to 2000 coffees per month. They cross at the break-even point (1000, 3500); to the right of it revenue lies above cost and the gap between the lines is shaded as profit.

Left of the intersection the cost line is on top → loss. Right of it, revenue is on top → profit.

Mixture Problems

Combining different concentrations or values

An investor has 10,000 EUR to split between bonds (4% return) and stocks (9% return). To earn 650 EUR 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 EUR in bonds, 5,000 EUR in stocks

Financial Growth Problems

Real-world application with compound interest

An investment of 5,000 EUR earns 8% annual interest compounded yearly.

  1. Write the formula for the amount after \(t\) years
  2. Calculate the value after 10 years
  3. When will it triple in value?

Cost-Profit Analysis

Combining algebraic techniques for Manufacturing

A company produces \(x\) units with:

  • Cost: \(C = 2x^2 + 100x + 5000\)
  • Revenue: \(R = 500x - 3x^2\)

Your tasks:

  1. How can you compute the profit?
  2. Factor the profit completely

Team Problem-Solving

Group Task

Work in groups on the following problem

A company produces two products:

  • Product A: Costs 15 EUR to make, sells for 25 EUR
  • Product B: Costs 20 EUR to make, sells for 35 EUR
  • Fixed costs: 5,000 EUR/month
  • Production capacity (the maximum possible output): 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?

Hint: Compare how much one more unit of A and one more unit of B add to the profit.

Challenge Problems

Test your skills by solving these in pairs

  1. If \(3^x + 3^{-x} = 4\), find \(9^x + 9^{-x}\)

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

  3. A bacteria culture triples every 4 hours. Starting with 500 bacteria:

    • When will there be 1 million bacteria?
    • Express the population after 1 day in scientific notation

Wrap-Up & Takeaways

Key Takeaways

Essential skills from Section 1

  • You can factor, simplify, and manipulate any expression type we studied
  • Translation from words to equations is systematic - use IDEA
  • Inequalities have special rules (flip when multiplying by negative!)
  • Break-even analysis is fundamental to business planning
  • Most importantly: You can combine techniques to solve complex problems

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

Study Strategy Going Forward

Success in mathematics requires consistent practice

Daily (10-15 minutes):

  • Review one concept from past sections
  • Solve 2-3 smaller problems
  • Check your solutions carefully

Do this in addition to our lecture and the problems we solve together here.

Final Assessment

Individual work

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

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

Next Session: Mock Exam 02

Session 01-07 is exam only - no lecture

  • 90 min + 5 min reading, 50 points, counts toward the 60%
  • Covers Section 01: factoring, exponents, logarithms, sets, absolute values, inequalities
  • FSP style: business problems whose parts build on each other
  • Calculator, drawing instruments, dictionary - no notes, formula sheets or electronic devices

Homework: Tasks 01-06 and FSP Training 01 (sidebar, Section 01 worksheets). Write down every step: the steps earn the points!