College Algebra Final Exam Review: Key Concepts and Practice Problems

Review of Algebra Fundamentals

Basic Definitions and Properties

This section covers foundational algebraic concepts, including definitions, properties of equality, and basic terminology essential for solving equations and inequalities.

• Horizontal Line: The slope of a horizontal line is 0.

• Vertical Line: The slope of a vertical line is undefined.

• Equation: A mathematical statement that asserts two expressions are equal.

• Expression: A representation involving numbers, variables, and operations, but no equality sign.

• Solution to an Equation: A value of the variable that makes the equation true.

• System of Equations: A set of two or more equations with the same variables.

• Consistent System: A system with at least one solution.

• Point of Intersection: The solution to a system of equations, represented as an ordered pair.

• Multiplication Property of Equality: If $a = b$, then $ac = bc$ for any $c$.

• Division Property of Equality: If $a = b$ and $c \neq 0$, then $\frac{a}{c} = \frac{b}{c}$.

• Inequality: A mathematical statement comparing two expressions using symbols such as $<, >, \leq, \geq$.

Equations and Inequalities

Solving Linear Equations and Inequalities

Linear equations and inequalities are solved by isolating the variable using algebraic operations. Solutions may be represented as numbers, intervals, or sets.

• Linear Equation Example: $7(x - 4) = 35$ Solution: $x = 9$

• Linear Inequality Example: $2x - 7 \leq 11$ Solution: $x \leq 9$

• Absolute Value Equation Example: $|x - 7| = 12$ Solution: $x = 19$ or $x = -5$

• Absolute Value Inequality Example: $|x - 7| \leq 12$ Solution: $-5 \leq x \leq 19$

• Radical Equation Example: $\sqrt{x - 1} = 4$ Solution: $x = 17$

Functions

Function Notation and Domain

Functions relate inputs to outputs according to a specific rule. The domain of a function is the set of all possible input values.

• Function Example: $f(x) = \frac{\sqrt{x}}{x - 100}$

• Domain Example: $[0, 100) \cup (100, \infty)$

• Interval Notation: Used to describe domains and solution sets, e.g., $(-\infty, 7) \cup (7, \infty)$.

Linear Functions

Graphing and Writing Equations of Lines

Linear functions are represented by straight lines and can be described by their slope and y-intercept. The equation of a line can be written in slope-intercept form: $y = mx + b$.

• Slope-Intercept Form: $y = mx + b$

• Finding Slope: Given two points $(x_1, y_1)$ and $(x_2, y_2)$, $m = \frac{y_2 - y_1}{x_2 - x_1}$

• Equation of a Line Through Two Points: Use the slope formula and one point to write the equation.

• Vertical Line Equation: $x = a$

• Perpendicular Lines: Slopes are negative reciprocals.

• Example: Find the equation of the line through $(1,2)$ and $(3,4)$: $m = \frac{4-2}{3-1} = 1$ Equation: $y = x + 1$

Polynomial and Rational Functions

Factoring and Simplifying Expressions

Polynomial expressions can be simplified and factored to solve equations and analyze functions. Rational expressions involve ratios of polynomials.

• Factoring Quadratics: $x^2 + 5x + 6 = (x + 2)(x + 3)$

• Factoring by Grouping: $x^2 - 2x - 15 = (x - 5)(x + 3)$

• Simplifying Rational Expressions: $\frac{x^2 - 4}{x^2 - 2x} = \frac{(x + 2)(x - 2)}{x(x - 2)} = \frac{x + 2}{x}$ for $x \neq 0, x \neq 2$

• Example: $\frac{2x + 4}{x + 2} = 2$ for $x \neq -2$

Radical and Exponential Expressions

Simplifying and Solving Radical Equations

Radical equations involve roots, and their solutions require isolating the radical and squaring both sides when necessary. Exponential expressions use properties of exponents for simplification.

• Radical Equation Example: $\sqrt{x + 1} = 4$ Solution: $x = 15$

• Exponent Properties: $a^m \cdot a^n = a^{m+n}$, $\frac{a^m}{a^n} = a^{m-n}$

• Example: $\frac{\sqrt{75}}{\sqrt{3}} = \sqrt{25} = 5$

Systems of Equations and Inequalities

Solving Systems Algebraically and Graphically

Systems of equations can be solved by substitution, elimination, or graphing. The solution is the point(s) where the equations intersect.

• Substitution Method: Solve one equation for a variable and substitute into the other.

• Elimination Method: Add or subtract equations to eliminate a variable.

• Example: Solve $\begin{cases} x + y = 5 \\ x - y = 1 \end{cases}$ Solution: $x = 3$, $y = 2$

• Possible Solutions: One solution, no solution, or infinitely many solutions.

Applications and Word Problems

Geometric and Real-World Applications

Algebraic methods are used to solve problems involving geometry, such as finding dimensions, using the Pythagorean theorem, and working with rectangles and triangles.

• Rectangle Area: $A = \text{Width} \times \text{Length}$

• Pythagorean Theorem: $a^2 + b^2 = c^2$

• Example: A rectangle has area 108, length $= w + 2$, width $= w$. Solve $w(w + 2) = 108$

• Right Triangle: If hypotenuse is 20 and one leg is $x$, other leg is $\sqrt{400 - x^2}$

Tables

Factoring Polynomials Table

This table summarizes the factoring of several polynomials as presented in the review problems.

Polynomial	Factored Form
$x^2 + 5x + 6$	$(x + 2)(x + 3)$
$x^2 - 2x - 15$	$(x - 5)(x + 3)$
$x^2 - 9$	$(x - 3)(x + 3)$
$x^2 + 2x - 24$	$(x + 6)(x - 4)$
$x^2 - 8x + 16$	$(x - 4)^2$
$x^2 - 49$	$(x - 7)(x + 7)$
$x^2 + 10x + 25$	$(x + 5)^2$
$x^2 - 100$	$(x - 10)(x + 10)$

Additional info:

• Some problems involve interval notation, radical simplification, and completing the square, which are standard topics in College Algebra.

• Word problems and geometric applications reinforce algebraic problem-solving skills.