Comprehensive Study Notes: Key Concepts in Beginning and Intermediate Algebra

Word Problems and Applications

Sales Tax, Total Cost, and Change

Understanding how to calculate total cost, including sales tax, and determining change is a fundamental application of arithmetic and algebra in real life.

• Step 1: Find the total cost of identical items Multiply the price per item by the number of items. Example: 3 pairs of socks at $1.85 each: $3 \times 1.85 = 5.55$

• Step 2: Find the subtotal before tax Add the cost of all items purchased. Example: $5.55 + 8.95 = 14.50$

• Step 3: Calculate the sales tax Convert the percent to a decimal and multiply by the subtotal. Example: $0.06 \times 14.50 = 0.87$

• Step 4: Find the total cost Add the sales tax to the subtotal. Example: $14.50 + 0.87 = 15.37$

• Step 5: Find the change Subtract the total cost from the amount paid. Example: $20.00 - 15.37 = 4.63$

Unit Rate and Cost per Ounce

Unit rates allow comparison of prices and quantities in different units.

• Step 1: Convert units if necessary 1 pound = 16 ounces.

• Step 2: Find cost per ounce Divide the price per pound by 16. Example: $2.80 \div 16 = 0.175$ dollars per ounce.

• Step 3: Convert to cents Multiply by 100. Example: $0.175 \times 100 = 17.5$ cents per ounce.

Geometry and Measurement

Hypotenuse of a Right Triangle

The Pythagorean Theorem relates the lengths of the sides of a right triangle.

• Step 1: Identify the legs Vertical leg: difference in y-values; Horizontal leg: difference in x-values.

• Step 2: Find the lengths For points (2,4), (2,−1), (0,−1): Vertical leg: $4 - (-1) = 5$ Horizontal leg: $2 - 0 = 2$

• Step 3: Apply the Pythagorean Theorem $a^2 + b^2 = c^2$ Substitute: $5^2 + 2^2 = c^2$ $25 + 4 = 29$ $c = \sqrt{29}$

Area of a Composite Figure

Composite figures are broken into simpler shapes for area calculation.

• Step 1: Decompose the figure Break into rectangles, circles, etc.

• Step 2: Use area formulas Rectangle: $A = lw$ Circle: $A = \pi r^2$ (use $\pi = 3.14$)

• Step 3: Substitute dimensions and add areas

Volume of a Cylinder

The volume of a cylinder is found using the formula:

• Formula: $V = \pi r^2 h$

• Step 1: Identify radius and height Radius is half the diameter.

• Step 2: Substitute values For diameter 10, $r = 5$; height $h = 4$ $V = 3.14 \times 5^2 \times 4$

Ratios, Proportions, and Percents

Solving Ratio Problems

Ratios compare quantities and can be used to divide groups proportionally.

• Step 1: Add ratio parts Residents:Visitors = 2:3; Total parts = $2 + 3 = 5$

• Step 2: Find value of one part Total people $\div$ total parts: $60 \div 5 = 12$

• Step 3: Find number in each group Visitors: $3 \times 12 = 36$

Percent Equations

Percent problems can be solved using equations.

• Step 1: Convert percent to decimal 75% = 0.75

• Step 2: Write and solve the equation $0.75x = 2.4$ Divide both sides by 0.75: $x = 2.4 \div 0.75 = 3.2$

Roots and Exponents

Square Roots and Cube Roots

Roots are the inverse operation of exponents.

• Square Root: A number that, when squared, gives the original number. Example: $\sqrt{64} = 8$ and $-8$ (since $8^2 = 64$ and $(-8)^2 = 64$)

• Cube Root: A number that, when cubed, gives the original number. Example: $\sqrt[3]{-64} = -4$ (since $(-4)^3 = -64$)

Exponents

Exponents represent repeated multiplication.

• Zero exponent: Any nonzero number to the zero power is 1. Example: $4^0 = 1$

• Negative exponent: $a^{-n} = \frac{1}{a^n}$ Example: $3^{-1} = \frac{1}{3}$

• Evaluate: $4^0 + 3^{-1} + 2^2 = 1 + \frac{1}{3} + 4 = 5 \frac{1}{3}$

Scientific Notation

Scientific notation expresses very large or small numbers as a product of a coefficient and a power of 10.

• Step 1: Multiply coefficients $(4.8 \times 10^{-10})(6 \times 10^{-6})$ Multiply: $4.8 \times 6 = 28.8$

• Step 2: Add exponents $10^{-10} \times 10^{-6} = 10^{-16}$

• Step 3: Write in proper form $28.8 \times 10^{-16} = 2.88 \times 10^{-15}$

Equations and Inequalities

Literal Equations

Literal equations involve solving for a variable in terms of others.

• Example (a): $x + c = d$ Subtract $c$ from both sides: $x = d - c$

• Example (b): $an = b$ Divide both sides by $a$: $n = \frac{b}{a}$

Solving Linear Inequalities

Solving inequalities is similar to solving equations, but the solution is a range of values.

• Example: $6x + 8 < 14$

• Subtract 8: $6x < 6$

• Divide by 6: $x < 1$

• Graph: Open circle at 1, shade to the left (values less than 1)

Order of Operations

Evaluating Expressions

Order of operations ensures consistent results in calculations.

• Step 1: Parentheses

• Step 2: Exponents

• Step 3: Multiplication/Division

• Step 4: Addition/Subtraction

• Example: $25 - [3^2 + 2(5 - 3)]$ First, $5 - 3 = 2$ Then, $3^2 = 9$ Then, $2 \times 2 = 4$ Add: $9 + 4 = 13$ Subtract: $25 - 13 = 12$

Functions and Graphing

Graphing Linear Equations

Linear equations can be represented as straight lines on the coordinate plane.

• Equation: $y = x - 2$

• Choose x-values and find y-values:

x	y
-1	-3
0	-2
1	-1
2	0

• Plot these points and draw a straight line through them.

Number Systems

Rational Numbers and the Number Line

Rational numbers can be written as fractions or decimals that terminate or repeat.

• Step 1: Convert all numbers to decimals

• Step 2: Order from least to greatest and plot on a number line

• Step 3: Identify rational numbers Numbers that can be written as $\frac{a}{b}$, where $a$ and $b$ are integers and $b \neq 0$.

Summary Table: Key Formulas and Concepts

Concept	Formula/Definition	Example
Sales Tax	$\text{Tax} = \text{Rate} \times \text{Subtotal}$	$0.06 \times 14.50 = 0.87$
Pythagorean Theorem	$a^2 + b^2 = c^2$	$5^2 + 2^2 = c^2$
Area of Rectangle	$A = lw$	$5 \times 3 = 15$
Area of Circle	$A = \pi r^2$	$3.14 \times 4^2 = 50.24$
Volume of Cylinder	$V = \pi r^2 h$	$3.14 \times 5^2 \times 4$
Percent Equation	$\text{Percent} \times x = \text{Amount}$	$0.75x = 2.4$
Scientific Notation	$a \times 10^n$	$2.88 \times 10^{-15}$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$	$3^{-1} = \frac{1}{3}$
Linear Function	$y = mx + b$	$y = x - 2$

Additional info: These notes synthesize and expand upon the provided material, ensuring each topic is self-contained and includes definitions, examples, and step-by-step procedures relevant to beginning and intermediate algebra students.