Section 02: Equations & Problem-Solving Strategies
10 minutes - individual work, then we review
Factor completely: \(x^6 - 7x^3 - 8\)
Simplify: \(\frac{(3x^{-2}y^3)^{-2} \cdot (2x^3y^{-1})^3}{6x^{-4}y^2}\)
If \(2^{x+1} + 2^x = 24\), find \(x\)
Rationalize and simplify: \(\frac{3}{\sqrt{5} + \sqrt{3}} - \frac{2}{\sqrt{5} - \sqrt{3}}\)
Present your solutions and we review together!
20 minutes - presentations and discussion
Remember: Discussing tasks helps solidify your own understanding!
A method to help you assess tasks
Today 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 necessaryly 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?
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?
Work in groups on the following problem
A company produces two products:
Work in groups on the following problem
Essential skills from today
Watch out for these!
Individual work
A small business has monthly costs of €3,000 plus €12 per unit produced. They sell each unit for €20.
Session 02-02: Systems of Equations
Preparation Tip
Review today’s equation-solving techniques - they’re the foundation for systems!
Session 02-01 - Equations & Inequalities | Dr. Nikolai Heinrichs & Dr. Tobias Vlćek | Home