Applications and More Algebra: Fundamental Concepts, Equations, Inequalities, and Sequences

1.1 Applications of Equations

Modeling and Translating Relationships

Mathematical modeling involves expressing real-world relationships using mathematical symbols and equations. This process is essential for solving practical problems in science, business, and everyday life.

• Example – Mixture Problem: A chemist must prepare 350 ml of a solution with two parts alcohol and three parts acid. Let n be the number of milliliters in each part. The equation is: $2n + 3n = 350$ $5n = 350$ $n = 70$ Amount of alcohol: $2n = 140$ ml Amount of acid: $3n = 210$ ml

• Business Applications:

  • Fixed cost: Costs independent of production level.

  • Variable cost: Costs dependent on output.

  • Total cost: $\text{variable cost} + \text{fixed cost}$

  • Total revenue: $(\text{price per unit}) \times (\text{number of units sold})$

  • Profit: $\text{total revenue} - \text{total cost}$

• Example – Profit Calculation: Variable cost per unit = $6$, fixed cost = $80,000$, selling price = $10$, desired profit = $60,000$. $10q - (6q + 80,000) = 60,000$ $4q = 140,000$ $q = 35,000$ Thus, 35,000 units must be sold to earn a profit of $60,000.

• Example – Investment Allocation: $10,000$ invested in A (6%) and B (5.75%), total interest $588.75$. $0.06x + 0.0575(10,000 - x) = 588.75$ $0.0025x = 13.75$ $x = 5,500$ $5,500$ at 6%, $4,500$ at 5.75%.

1.2 Linear Inequalities

Understanding and Solving Inequalities

An inequality compares two values, showing if one is less than, greater than, or equal to another. Linear inequalities involve variables and can be solved similarly to equations, but with special rules for operations involving negative numbers.

• Relative Positions: For points $a$ and $b$ on the real number line, $a = b$, $a < b$, or $a > b$.

• Interval Notation: If $a < x < b$, $x$ is between $a$ and $b$.

Rules for Inequalities

• If $a < b$, then $a + c < b + c$ and $a - c < b - c$.

• If $a < b$ and $c > 0$, then $ac < bc$ and $\frac{a}{c} < \frac{b}{c}$.

• If $a < b$ and $c < 0$, then $ac > bc$ and $\frac{a}{c} > \frac{b}{c}$.

• If $a < b$ and $a = c$, then $c < b$.

• If $0 < a < b$, then $\frac{1}{a} > \frac{1}{b}$.

• If $0 < a < b$ and $n > 0$, then $a^n < b^n$. If $0 < a < b$, then $\sqrt{a} < \sqrt{b}$.

Solving Linear Inequalities

• A linear inequality in $x$ can be written as $ax + b < 0$, where $a \neq 0$.

• To solve, isolate $x$ using equivalent transformations.

Example: Solve $2(x - 3) < 4$ $2x - 6 < 4$ $2x < 10$ $x < 5$ Solution: $x < 5$ Example: Solve $\frac{3}{2}(s-2) + 1 > -2(s-4)$ $3(s-2) + 2 > -4(s-4)$ $3s - 6 + 2 > -4s + 16$ $3s + 4s > 16 + 6 - 2$ $7s > 20$ $s > \frac{20}{7}$ Solution: $s > \frac{20}{7}$

1.3 Applications of Inequalities

Solving Word Problems with Inequalities

• Example – Profit Condition: For a company making heaters, cost per heater = $21$, fixed cost = $70,000$, selling price = $35$. $35q - (21q + 70,000) > 0$ $14q > 70,000$ $q > 5,000$ At least 5,001 heaters must be sold to earn a profit.

1.4 Absolute Value

Definition and Properties

The absolute value of a real number $x$, denoted $|x|$, is its distance from 0 on the real number line.

• Definition: $|x| = \begin{cases} x, & x \geq 0 \\ -x, & x < 0 \end{cases}$

Solving Absolute Value Equations

• Example: Solve $|x-3| = 2$ $x-3 = 2$ or $x-3 = -2$ $x = 5$ or $x = 1$

• Example: Solve $|7-3x| = 5$ $7-3x = 5$ or $7-3x = -5$ $x = \frac{2}{3}$ or $x = 4$

Absolute Value Inequalities

Inequality	Solution
$|x| < a$	$-a < x < a$
$|x| > a$	$x < -a$ or $x > a$
$|x| \leq a$	$-a \leq x \leq a$
$|x| \geq a$	$x \leq -a$ or $x \geq a$

Properties of Absolute Value

• $|-7| = 7$

• $|4 - 12| = 8$

• $|3 - 7| = 4$

• $|-3| + |7| = 10$

• $|3| + |-7| = 10$

• $|5 - 2| + |3| = 3 + 3 = 6$

1.5 Summation Notation

Definition and Evaluation

The sum of numbers $a_i$ for $i$ from $m$ to $n$ is denoted $\sum_{i=m}^n a_i$.

• Example: $\sum_{n=3}^7 (5n-2)$ $= (5 \times 3 - 2) + (5 \times 4 - 2) + (5 \times 5 - 2) + (5 \times 6 - 2) + (5 \times 7 - 2)$ $= 13 + 18 + 23 + 28 + 33 = 115$

• Example: $\sum_{i=1}^6 (i^2 + 1)$ $= (1^2 + 1) + (2^2 + 1) + ... + (6^2 + 1)$ $= 2 + 5 + 10 + 17 + 26 + 37 = 97$

• Formula for sum of first $n$ integers: $\sum_{i=1}^n i = \frac{n(n+1)}{2}$

Properties of Summation

• $\sum_{j=30}^{100} 4 = 4 \times 71 = 284$

• $\sum_{k=1}^{200} 9k^2 = 9 \sum_{k=1}^{200} k^2 = 9 \left( \frac{200 \cdot 201 \cdot 401}{6} \right) = 24,180,300$

1.6 Sequences

Definitions and Types

A sequence of length $n$ assigns to each element of the set $\{1, 2, ..., n\}$ exactly one real number. A finite sequence has a finite number of terms; an infinite sequence assigns a real number to each positive integer.

• Example: For $a_k = 2k^2 + 3k + 1$, first four terms: $a_1 = 6$, $a_2 = 15$, $a_3 = 28$, $a_4 = 45$

• Example: For $e_k = \frac{(k+1)^k}{k^k}$, first four terms: $e_1 = 2$, $e_2 = \frac{9}{4}$, $e_3 = \frac{64}{27}$, $e_4 = \frac{625}{256}$

Recursively Defined Sequences

• A sequence defined in terms of previous terms and initial values.

• Example – Fibonacci Sequence: $F_1 = 1$, $F_2 = 1$, $F_{k+2} = F_{k+1} + F_k$ First ten terms: $1, 1, 2, 3, 5, 8, 13, 21, 34, 55$

Arithmetic Sequence & Geometric Sequence

Definitions and Examples

• Arithmetic Sequence: $b_1 = a$, $b_{k+1} = d + b_k$ Common difference $d = b_{k+1} - b_k$

• Example: $a = 1.5$, $d = 0.7$, sequence: $1.5, 2.2, 2.9, 3.6, 4.3, 5.0$

• Geometric Sequence: $c_1 = a$, $c_{k+1} = c_k r$ Common ratio $r = \frac{c_{k+1}}{c_k}$

• Example: $a = \sqrt{2}$, $r = 1/2$, sequence: $\sqrt{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{4}, \frac{\sqrt{2}}{8}, \frac{\sqrt{2}}{16}$

Sums of Sequences

• Sum of arithmetic sequence: $s_n = \frac{n}{2}((n-1)d + 2a)$

• Sum of geometric sequence ($r \neq 1$): $s_n = a \frac{1 - r^n}{1 - r}$

• Sum of infinite geometric sequence ($|r| < 1$): $\sum_{n=1}^\infty ar^{n-1} = \frac{a}{1 - r}$

Example – Repeating Decimals as Geometric Series

• Is $0.151515... = \frac{5}{33}$? $0.151515... = 0.15 + 0.0015 + 0.000015 + ...$ $a = 0.15$, $r = 0.01$ $0.151515... = \frac{0.15}{1 - 0.01} = \frac{0.15}{0.99} = \frac{15}{99} = \frac{5}{33}$

Additional info: This summary covers the main algebraic concepts, equations, inequalities, absolute value, summation notation, and sequences as presented in the provided lecture notes. All formulas and examples are expanded for clarity and exam preparation.