Section 01: Mathematical Foundations & Algebra
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!
Comprehensive discussions
Today we see how all pieces fit together!
By the end of this session, you can:
Essential foundations
Number hierarchy: \(\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{I} \subset \mathbb{R}\)
Set operations:
Quick practice:
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 \(\frac{(3x^2)^3}{9x^4}\)
Your factoring toolbox
Quick practice: Factor \(x^4 - 16\)
Advanced operations
Radicals: extract perfect powers; rationalize with conjugates: \((a + \sqrt{b})(a - \sqrt{b}) = a^2 - b\)
Logarithms:
Quick practice: Solve \(2^x = 10\)
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
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: \(x^3 + 2x^2 - 9x - 18\)
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: \(\frac{2}{\sqrt{6} - \sqrt{2}}\)
A method to help you assess tasks
Now we apply IDEA to translating word problems into equations and inequalities!
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” → \(0.3S\) |
Key terms you’ll encounter frequently
Always define your variables clearly before translating!
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”
A systematic approach
Let’s work through this together
Solve: \(\frac{2x - 1}{3} + \frac{x + 2}{4} = 5\)
When things aren’t necessarily equal
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?
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?
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?
Left of the intersection the cost line is on top → loss. Right of it, revenue is on top → profit.
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?
Real-world application with compound interest
An investment of €5,000 earns 8% annual interest compounded yearly.
Combining algebraic techniques for Manufacturing
A company produces \(x\) units with:
Your tasks:
Work in groups on the following problem
A company produces two products:
Work in groups on the following problem
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:
Essential skills from Section 1
Watch out for these!
Success in mathematics requires consistent practice
Daily (10-15 minutes):
Do this in addition to our lecture and the problems we solve together here.
Individual work
A small business has monthly costs of €3,000 plus €12 per unit produced. They sell each unit for €20.
Session 02-01: Systems of Linear Equations
Homework Assignment
Complete Tasks 01-06! Today’s equation-solving techniques are the foundation for systems.
Session 01-06 - Synthesis & Linear Equations | Dr. Nikolai Heinrichs & Dr. Tobias Vlćek | Home