College Algebra Final Exam Review: Key Concepts and Practice Problems

Linear Equations and Inequalities in One Variable

Evaluating and Simplifying Expressions

Understanding how to evaluate algebraic expressions and simplify them is foundational in algebra. This includes substituting values for variables and performing arithmetic operations according to the order of operations.

• Evaluate: Substitute the given value for the variable and compute the result.

• Simplify: Combine like terms and use the distributive property as needed.

• Example: Evaluate $4x^2 + 2x$ for $x = 2$.

Scientific Notation

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

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

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

Solving Linear Equations

Solving linear equations involves isolating the variable on one side of the equation.

• Steps:

  1. Combine like terms on each side.

  2. Use addition or subtraction to get all variable terms on one side and constants on the other.

  3. Divide or multiply to solve for the variable.

• Example: Solve $3x + 5 = 11$.

Solving Inequalities

Similar to equations, but remember to reverse the inequality sign when multiplying or dividing by a negative number.

• Example: Solve $2x - 3 > 7$.

Exponents and Polynomials

Properties of Exponents

Exponents follow specific rules that simplify expressions involving powers.

• 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: Simplify $x^3 \cdot x^4$.

Factoring Polynomials

Factoring is the process of writing a polynomial as a product of its factors.

• Common Methods:

  • Factoring out the greatest common factor (GCF)

  • Factoring trinomials

  • Difference of squares: $a^2 - b^2 = (a - b)(a + b)$

• Example: Factor $x^2 - 9$.

Rational Expressions

Simplifying Rational Expressions

Rational expressions are fractions where the numerator and/or denominator are polynomials. Simplify by factoring and reducing common factors.

• Example: Simplify $\frac{x^2 - 9}{x - 3}$.

Operations with Rational Expressions

• Addition/Subtraction: Find a common denominator before combining.

• Multiplication/Division: Multiply numerators and denominators; for division, multiply by the reciprocal.

• Example: $\frac{3}{x} + \frac{2}{x} = \frac{5}{x}$

Linear Equations in Two Variables

Writing Equations of Lines

The equation of a line can be written in several forms, most commonly slope-intercept form.

• Slope-Intercept Form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Point-Slope Form: $y - y_1 = m(x - x_1)$

• Example: Write the equation of the line passing through $(1,2)$ with slope $3$.

Graphing Linear Equations

Plot points and use the slope to draw the line.

• Steps:

  1. Plot the y-intercept.

  2. Use the slope to find another point.

  3. Draw the line through the points.

Systems of Linear Equations

Solving Systems of Equations

Systems can be solved by graphing, substitution, or elimination.

• Graphing: Plot both equations and find the intersection point.

• 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 $x - y = 1$.

Applications and Word Problems

Translating Words to Equations

Many real-world problems can be modeled with algebraic equations.

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

Mixture and Ticket Problems

Set up equations based on totals and relationships described in the problem.

• Example: Tickets for an amusement park cost $10 each for adults and $5 each for children. If 150 tickets were sold for a total of $1050, how many tickets were sold for children and adults?

Table: Properties of Exponents

Property	Rule	Example
Product of Powers	$a^m \cdot a^n = a^{m+n}$	$x^2 \cdot x^3 = x^5$
Quotient of Powers	$\frac{a^m}{a^n} = a^{m-n}$	$x^5 / x^2 = x^3$
Power of a Power	$(a^m)^n = a^{mn}$	$(x^2)^3 = x^6$
Power of a Product	$(ab)^n = a^n b^n$	$(2x)^3 = 8x^3$
Zero Exponent	$a^0 = 1$ (if $a \neq 0$)	$5^0 = 1$