Applications and More Algebra ?

Equations and Inequalities

Equations: Mathematical statements asserting that two expressions are equal. They typically involve one or more variables and are used to find a specific value or set of values that makes the statement true. Equations are fundamental for modeling situations where quantities are balanced or have an exact relationship.
- Example: $2x + 5 = 11$ is an equation where a specific value of $x$ (in this case, $x=3$ ) satisfies the equality.
Inequalities: Mathematical statements that compare two expressions using inequality symbols, indicating that one quantity is less than ( < ), greater than ( > ), less than or equal to ( $\le$ ), or greater than or equal to ( $\ge$ ) another. Unlike equations that yield specific solutions, inequalities often describe a range of possible values.
- Example: 2x + 5 < 11 is an inequality, whose solution is x < 3 , representing all numbers less than 3.

Applications of Equations

Modeling: The process of translating real-world problems and stated relationships into mathematical equations. This involves:
1. Understanding the problem: Identify knowns, unknowns, and the goal.
2. Assigning variables: Represent unknown quantities with variables (e.g., $x, q, r$ ).
3. Formulating the equation: Write down mathematical expressions based on the relationships described in the problem.
4. Solving the equation: Use algebraic techniques to find the value(s) of the variable(s).
5. Interpreting the solution: Relate the mathematical solution back to the context of the original problem.
Mixture Problem Example (Detailed): To prepare $350 \text{ ml}$ of a solution where alcohol and acid are mixed in a ratio of $2$ parts alcohol to $3$ parts acid:
1. Define a unit: Let $n$ represent the volume (in ml) of one part.
2. Express component volumes: Alcohol volume = $2n$ . Acid volume = $3n$ .
3. Formulate equation: The total volume is the sum of component volumes: $2n + 3n = 350$ .
4. Solve: $5n = 350 \Rightarrow n = 70 \text{ ml}$ .
5. Calculate component amounts: Alcohol: $2 \times 70 = 140 \text{ ml}$ . Acid: $3 \times 70 = 210 \text{ ml}$ .
Business Terms (Detailed):
- Fixed Cost (FC): Costs that do not change with the level of production or sales. These are incurred even if nothing is produced. Examples include rent, insurance, and salaries of administrative staff.
- Variable Cost (VC): Costs that vary directly with the number of units produced. As production increases, variable costs increase. Examples include raw materials, direct labor, and sales commissions.
- Total Cost (TC): The sum of all costs incurred in producing a certain quantity of goods. $\text{TC} = \text{FC} + \text{VC}$ . Often, variable cost is expressed as $VC = vq$ where $v$ is the variable cost per unit and $q$ is the quantity produced. So, $\text{TC} = \text{FC} + vq$ .
- Total Revenue (TR): The total amount of money received from selling goods or services. $\text{TR} = (\text{Price per unit}, p) \times (\text{Number of units sold}, q) = pq$ .
- Profit (\varPi): The financial gain realized when the amount of revenue gained from a business activity exceeds the expenses, costs, and taxes involved. $\varPi = \text{TR} - \text{TC}$ .
- Breakeven Point: The point at which total cost and total revenue are equal, meaning there is no net loss or gain ( $\varPi = 0$ ). This is found by setting $\text{TR} = \text{TC}$ .
Profit Example (Detailed): A company has a variable cost $\$6/unit$ , fixed cost $\$80,000$ , and sells its product for $\$10/unit$ . To achieve a profit of $\$60,000$ :
1. Identify formulas: $\varPi = \text{TR} - \text{TC}$ , where $\text{TR} = 10q$ and $\text{TC} = 6q + 80,000$ .
2. Set up the equation: $60,000 = 10q - (6q + 80,000)$ .
3. Solve for $q$ :
  $60,000 = 10q - 6q - 80,000$
  $60,000 = 4q - 80,000$
  $140,000 = 4q$
  $q = 35,000 \text{ units}$ .
  The company needs to sell $35,000$ units to achieve a $\$60,000$ profit.
Investment Example (Detailed): Investing a total of $\$10,000$ in two ventures, A and B. Venture A offers a 6% annual return, and Venture B offers a 5.75% annual return. The total annual earnings are $\$588.75$ .
1. Assign variables: Let $x$ be the amount invested in Venture A. Then the amount invested in Venture B is $10,000 - x$ .
2. Formulate equation: The total earnings are the sum of earnings from each venture:
  $0.06x + 0.0575(10,000 - x) = 588.75$
3. Solve for $x$ :
  $0.06x + 575 - 0.0575x = 588.75$
  $0.0025x = 588.75 - 575$
  $0.0025x = 13.75$
  $x = \frac{13.75}{0.0025}$
  $x = 5500$
4. Calculate amounts: $\$5500$ in Venture A, and $10,000 - 5500 = \$4500$ in Venture B.
Compound Interest (Basic and General): The future value ( $FV$ ) of an investment with simple compounding can be calculated. For a future value of $\$1,102,500$ needed in $2$ years from a $\$1,000,000$ initial investment:
1. Formula (general): $FV = P(1+r)^t$ , where $P$ is the principal, $r$ is the annual interest rate, and $t$ is the number of years.
2. Set up equation: $1,102,500 = 1,000,000(1+r)^2$ .
3. Solve for $r$ :
  $\frac{1,102,500}{1,000,000} = (1+r)^2$
  $1.1025 = (1+r)^2$
  $\sqrt{1.1025} = 1+r$
  $1.05 = 1+r$
  $r = 0.05$
  The required annual rate is $0.05$ (or 5%).

Linear Inequalities

Inequality Symbols and Sense:
- < : Less than (e.g., x < 5 )
- > : Greater than (e.g., x > 5 )
- $\le$ : Less than or equal to (e.g., $x \le 5$ )
- $\ge$ : Greater than or equal to (e.g., $x \ge 5$ )
  The "sense" or "direction" of an inequality refers to which side is larger or smaller.
Solving Inequalities: The goal is to isolate the variable, similar to solving equations.
- General Rules:
1. Adding or subtracting the same quantity from both sides does not change the sense of the inequality.
2. Multiplying or dividing both sides by a positive number does not change the sense of the inequality.
3. Multiplying or dividing both sides by a negative number reverses the sense of the inequality. This is a crucial difference from solving equations.
- Graphical Representation: Solutions to linear inequalities can be represented on a number line. An open circle indicates that the endpoint is not included ( < or > ), while a closed circle indicates it is included ( $\le$ or $\ge$ ).
Interval Notation: A concise way to describe the set of real numbers that satisfy an inequality.
- $(a, b)$ : Open interval from $a$ to $b$ . It includes all numbers $x$ such that a < x < b . Endpoints are not included.
- $[a, b]$ : Closed interval from $a$ to $b$ . It includes all numbers $x$ such that $a \le x \le b$ . Endpoints are included.
- $(a, b]$ : Half-open/half-closed interval. Includes $x$ such that a < x \le b .
- $[a, b)$ : Half-open/half-closed interval. Includes $x$ such that a \le x < b .
- $(-\infty, b)$ : All numbers $x$ such that x < b . The symbol $-\infty$ means "negative infinity" and always uses a parenthesis.
- $(a, \infty)$ : All numbers $x$ such that x > a . The symbol $\infty$ means "positive infinity" and always uses a parenthesis.
- $(-\infty, b]$ : All numbers $x$ such that $x \le b$ .
- $[a, \infty)$ : All numbers $x$ such that $x \ge a$ .
Solving Example (Detailed): Solve the inequality 2(x-3) < 4 and express the solution in interval notation.
1. Distribute: 2x - 6 < 4
2. Add 6 to both sides: 2x < 10
3. Divide by 2 (positive number, no sense change): x < 5
4. Interval Notation: The solution set includes all numbers less than 5, represented as $(-\infty, 5)$ .

Formula Sheet

Business Formulas

Total Cost (TC): $\text{TC} = \text{Fixed Cost (FC)} + \text{Variable Cost (VC)}$
If variable cost is per unit ( $v$ ) and quantity is $q$ : $\text{TC} = \text{FC} + vq$
Total Revenue (TR): $\text{TR} = \text{Price per unit (p)} \times \text{Quantity sold (q)} = pq$
Profit (\varPi): $\varPi = \text{Total Revenue (TR)} - \text{Total Cost (TC)}$
Breakeven Point: $\text{TR} = \text{TC}$ (i.e., $\varPi = 0$ )

Investment Formulas

Simple Compound Interest Future Value: $FV = P(1+r)^t$
- $FV$ = Future Value
- $P$ = Principal (initial investment)
- $r$ = Annual interest rate (as a decimal)
- $t$ =