Session 01-06 - Synthesis & Linear Equations

Section 01: Mathematical Foundations & Algebra

Author

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz - 10 Minutes

Quick Skills Check

Test your readiness

Simplify: $(2x^3)^2 \cdot x^{-5}$
Factor: $x^2 - 5x - 14$
Evaluate: $\log_2(32)$
Simplify: $\sqrt{48}$
Express in scientific notation: $0.000425$

. . .

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

Homework Discussion - 15 Minutes

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

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: $\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{I} \subset \mathbb{R}$

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

Your factoring toolbox

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

Individual Exercise

10 minutes - repeat the basics

Simplify: $\frac{x^3 \cdot x^{-5}}{x^{-3}}$
Factor completely: $3x^2 - 27$
Rationalize: $\frac{6}{\sqrt{3}}$
Solve: $|3x - 6| = 9$
Evaluate: $\log_3(81) - \log_3(3)$
Express in scientific notation: 45,600,000

Break - 10 Minutes

Practice Block 2 - Fundamentals

Pair Exercise

15 minutes - work together

Factor using substitution: $x^4 - 5x^2 + 4$
Simplify: $\sqrt{75} + \sqrt{48} - \sqrt{27}$
If $2^x = 3$ and $3^y = 5$, find $2^{xy}$
Rationalize: $\frac{4}{3 - \sqrt{5}}$
Factor by grouping: $x^3 + 2x^2 - 9x - 18$

Mixed Technique Problems

Combine multiple methods

Factor: $8x^3 - 27y^3$
Simplify: $\frac{x^3 - 8}{x^2 - 4} \cdot \frac{x + 2}{x^2 + 2x + 4}$
If $|2x - 5| < 7$, find the solution interval
Rationalize: $\frac{2}{\sqrt{6} - \sqrt{2}}$

From 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” → $x ≤ 100$
“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:

“Seven more than twice a number equals 31”
“The quotient of a number and 4, decreased by 3, is 12”
“40% of a number increased by 25 equals the number itself”

Linear Equations & Inequalities

Solving Multi-Step Equations

A systematic approach

Clear fractions: Multiply by LCD
Expand: Remove parentheses using distributive property
Collect terms: Variables on one side, constants on other
Isolate variable: Divide by coefficient
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

When things aren’t necessarily 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

Coffee Break - 15 Minutes

Guided Practice

Individual Exercises

Work independently, then we’ll discuss

To equation: “Three times a number decreased by 7 equals 14”
Solve: $3(2x - 4) = 2(x + 5)$
Solve the inequality: $-3x + 7 < 16$
A taxi charges €3.50 base fare plus €1.20 per km. If a ride costs €15.50, how far was it?
A store offers 30% discount. After discount, an item costs €42. 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/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

Break-Even, Seen Graphically

. . .

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 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

Financial Growth Problems

Real-world application with compound interest

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

Write the formula for the amount after $t$ years
Calculate the value after 10 years
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:

How can you compute the profit?
Factor the profit completely

Collaborative Problem-Solving - 30 Minutes

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

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

Challenge Problems

Test your skills by solving these in pairs

If $3^x + 3^{-x} = 4$, find $9^x + 9^{-x}$
Factor completely: $x^6 - 7x^3 - 8$
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 & Key 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 plus €12 per unit produced. They sell each unit for €20.

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

Next Session Preview

Session 02-01: Systems of Linear Equations

Solving systems by substitution and elimination
Business applications with multiple constraints
Extending to 3×3 systems with Gaussian elimination

. . .

Homework Assignment

Complete Tasks 01-06! Today’s equation-solving techniques are the foundation for systems.

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	\(a^{-n} = \frac{1}{a^n}\)	\(x^{-2} = \frac{1}{x^2}\)