College Algebra Final Exam Review: Key Concepts and Practice

Linear Equations and Inequalities in One Variable

Evaluating and Solving Linear Equations

Linear equations in one variable are equations that can be written in the form ax + b = c. Solving these equations involves isolating the variable.

• Evaluating Expressions: Substitute the given value for the variable and perform arithmetic operations.

• Solving for a Variable: Use inverse operations to isolate the variable on one side of the equation.

• Example: Solve $2x + 3 = 11$ Subtract 3 from both sides: $2x = 8$ Divide by 2: $x = 4$

Scientific Notation

Scientific notation expresses numbers as a product of a number between 1 and 10 and a power of 10.

• Format: $a \times 10^n$ where $1 \leq a < 10$ and $n$ is an integer.

• Example: $0.0005 = 5 \times 10^{-4}$

Simplifying Expressions

Simplifying involves combining like terms and applying arithmetic operations.

• Example: $3x + 2x = 5x$

• Order of Operations: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction (PEMDAS)

Linear Equations in Two Variables

Writing Equations of Lines

The equation of a line can be written in slope-intercept form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Given Two Points: Use the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$ to find the slope.

• Example: Find the equation of the line passing through (1,2) and (3,4).

Graphing Linear Equations

To graph a linear equation, plot the y-intercept and use the slope to find another point.

• Steps:

  1. Plot the y-intercept ($b$).

  2. From the y-intercept, use the slope ($m$) to find another point.

  3. Draw a straight line through the points.

Systems of Linear Equations

Solving Systems of Equations

A system of linear equations consists of two or more equations with the same variables. The solution is the point(s) where the equations intersect.

• Methods:

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

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

• Example: Solve $x + y = 3$ and $2x - y = 2$.

  1. Solve the first equation for $y$: $y = 3 - x$

  2. Substitute into the second: $2x - (3 - x) = 2$

  3. Simplify: $2x - 3 + x = 2 \Rightarrow 3x = 5 \Rightarrow x = \frac{5}{3}$

  4. Find $y$: $y = 3 - \frac{5}{3} = \frac{4}{3}$

Exponents and Polynomials

Properties of Exponents

Exponents indicate repeated multiplication. Key properties help simplify expressions.

• Product Rule: $a^m \cdot a^n = a^{m+n}$

• Quotient Rule: $\frac{a^m}{a^n} = a^{m-n}$

• Power Rule: $(a^m)^n = a^{mn}$

• Example: $2^3 \cdot 2^4 = 2^{3+4} = 2^7$

Polynomials and Factoring

A polynomial is an expression consisting of variables and coefficients, involving only addition, subtraction, and multiplication.

• Factoring: Expressing a polynomial as a product of its factors.

• Example: Factor $x^2 - 9 = (x - 3)(x + 3)$

Rational Expressions

Simplifying Rational Expressions

Rational expressions are fractions with polynomials in the numerator and denominator. Simplification involves factoring and reducing common factors.

• Example: $\frac{x^2 - 9}{x^2 - 3x} = \frac{(x - 3)(x + 3)}{x(x - 3)} = \frac{x + 3}{x}$

Solving Equations Involving Rational Expressions

To solve, find a common denominator, clear fractions, and solve the resulting equation.

• Example: $\frac{3}{x} = \frac{1}{2}$ Multiply both sides by $2x$: $6 = x$

Word Problems and Applications

Translating and Solving Word Problems

Word problems require translating real-world scenarios into algebraic equations.

• Example: "The length of a rectangle is 2 m more than the width. The area is 48 m2. Find the length and width."

  1. Let width = $x$, length = $x + 2$

  2. Area: $x(x + 2) = 48$

  3. Solve: $x^2 + 2x - 48 = 0$

Additional info:

• Some questions involve graphing and interpreting linear equations, which are foundational for understanding functions and systems.

• Problems cover a range of College Algebra topics, including exponents, polynomials, rational expressions, and applications.