College Algebra Final Exam Review: Key Concepts and Practice

Equations & Inequalities

Evaluating Expressions

Evaluating algebraic expressions involves substituting values for variables and performing arithmetic operations according to the order of operations.

• Key Point: Substitute the given values and simplify step by step.

• Example: If $x = 2$, $y = 3$, then $2x + 3y = 2(2) + 3(3) = 4 + 9 = 13$.

Scientific Notation

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

• Key Point: $a imes 10^n$, where $1 ext{ ≤ } a < 10$ and $n$ is an integer.

• Example: $4500 = 4.5 imes 10^3$

Simplifying Expressions

Simplifying involves combining like terms and applying algebraic properties.

• Key Point: Use distributive, associative, and commutative properties.

• Example: $2(x + 3) = 2x + 6$

Equations & Inequalities

Solving Linear Equations

Linear equations are solved by isolating the variable using inverse operations.

• Key Point: Perform the same operation on both sides to maintain equality.

• Example: $2x + 3 = 7 \Rightarrow 2x = 4 \Rightarrow x = 2$

Solving Quadratic Equations

Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.

• Quadratic Formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Example: $x^2 - 5x + 6 = 0 \Rightarrow (x-2)(x-3) = 0 \Rightarrow x = 2, 3$

Solving Inequalities

Inequalities are solved similarly to equations, but multiplying or dividing by a negative reverses the inequality sign.

• Key Point: Always check for sign reversal when multiplying/dividing by negatives.

• Example: $-2x > 6 \Rightarrow x < -3$

Functions

Evaluating and Graphing Functions

A function relates each input to exactly one output. Graphing functions involves plotting points that satisfy the function's equation.

• Key Point: The graph of $y = f(x)$ shows all $(x, y)$ pairs where $y$ is the output for input $x$.

• Example: For $f(x) = 2x + 1$, plot points for $x = -1, 0, 1$.

Writing Equations of Lines

The equation of a line can be written in slope-intercept form ($y = mx + b$) or point-slope form ($y - y_1 = m(x - x_1)$).

• Slope-intercept form: $y = mx + b$

• Point-slope form: $y - y_1 = m(x - x_1)$

• Example: Line through $(1,2)$ with slope $3$:$y - 2 = 3(x - 1)$

Polynomial & Rational Expressions

Factoring Polynomials

Factoring is expressing a polynomial as a product of its factors.

• Key Point: Look for common factors, differences of squares, or use grouping.

• Example: $x^2 - 9 = (x - 3)(x + 3)$

Simplifying Rational Expressions

Rational expressions are simplified by factoring numerator and denominator and canceling common factors.

• Key Point: Always check for restrictions on the variable (values that make denominator zero).

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

Systems of Equations

Solving Systems of Linear Equations

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

• Key Point: The solution is the point(s) where the equations intersect.

• Example: $x + y = 5$, $x - y = 1$; add to get $2x = 6 \Rightarrow x = 3$, then $y = 2$.

Applications

Word Problems

Translating word problems into algebraic equations is a key skill in College Algebra.

• Key Point: Define variables, write equations, and solve for unknowns.

• Example: "The length of a rectangle is 2 more than the width. The area is 48." Let $w$ be width, then length is $w+2$. $w(w+2) = 48$.

Exponents & Radicals

Simplifying Expressions with Exponents

Apply the laws of exponents to 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: $x^2 \cdot x^3 = x^{2+3} = x^5$

Simplifying Radicals

Radicals are simplified by factoring out perfect squares and applying the properties of square roots.

• Key Point: $\sqrt{a^2} = a$ for $a \geq 0$

• Example: $\sqrt{18} = \sqrt{9 \cdot 2} = 3\sqrt{2}$

HTML Table: Laws of Exponents

Law	Formula	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 $a \neq 0$)	$5^0 = 1$
Negative Exponent	$a^{-n} = \frac{1}{a^n}$	$x^{-2} = \frac{1}{x^2}$

Additional info:

• Some questions involve graphing and writing equations of lines, which are foundational in College Algebra.

• Word problems include systems of equations and applications of algebraic concepts.

• All topics covered are directly relevant to College Algebra, including equations, inequalities, functions, exponents, radicals, and systems.