College Algebra Study Guide: Linear, Rational, Quadratic Equations, Complex Numbers, and Inequalities

Linear and Rational Equations

Solving Linear Equations

Linear equations are equations of the form ax + b = 0, where a and b are constants. Solving linear equations involves isolating the variable on one side of the equation.

• Key Point 1: Combine like terms and use inverse operations to solve for the variable.

• Key Point 2: Check your solution by substituting back into the original equation or using a graphing calculator.

• Example: Solve $2x + 2 = 3x - 3$ $2x + 2 = 3x - 3$ $2x - 3x = -3 - 2$ $-x = -5$ $x = 5$

Solving Rational Equations

Rational equations contain fractions with variables in the denominator. To solve, clear denominators by multiplying both sides by the least common denominator (LCD).

• Key Point 1: Identify the LCD and multiply both sides to eliminate fractions.

• Key Point 2: Solve the resulting linear equation.

• Example: Solve $\frac{2x}{14} - \frac{2}{14} = \frac{7x}{119}$ Multiply both sides by 119: $28x - 4x = 7x$ $24x = 7x$ $17x = 119$ $x = 7$

Models and Applications

Solving Word Problems

Word problems require translating real-world situations into algebraic equations. Follow a systematic approach to identify variables, set up equations, and solve.

• Key Point 1: Define variables and write equations based on the problem statement.

• Key Point 2: Solve the equation and interpret the solution in context.

• Example: If $1800x + x = 252$, solve for $x$: $1800x + x = 252$ $1801x = 252$ $x = \frac{252}{1801}$

Complex Numbers

Definition and Properties

Complex numbers are numbers of the form a + bi, where a is the real part and b is the imaginary part. The imaginary unit i is defined as $i = \sqrt{-1}$.

• Key Point 1: The real part is a, and the imaginary part is b.

• Key Point 2: Standard form: $a + bi$

• Example: $3 + 4i$ is a complex number.

Addition, Subtraction, Multiplication, and Division

Operations with complex numbers follow algebraic rules, with $i^2 = -1$.

• Addition/Subtraction: Add/subtract real and imaginary parts separately.

• Multiplication: Use distributive property and substitute $i^2 = -1$.

• Division: Multiply numerator and denominator by the conjugate of the denominator.

• Example: $(3 + 4i) + (2 - 5i) = 5 - i$

Quadratic Equations

Definition and Methods of Solution

A quadratic equation is an equation of the form ax^2 + bx + c = 0. Solutions can be found by factoring, using the square root property, or the quadratic formula.

• Key Point 1: Factoring: Write the equation as a product of binomials and set each factor to zero.

• Key Point 2: Square Root Property: If $x^2 = a$, then $x = \pm \sqrt{a}$.

• Key Point 3: Quadratic Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Example: Solve $x^2 - 4x + 4 = 0$ $(x - 2)^2 = 0$ $x = 2$

Solving Quadratic Equations by Factoring

Factoring is a method where the equation is rewritten as a product of two binomials.

• Example: $2x^2 + 2x = 0$ $2x(x + 1) = 0$ $x = 0$ or $x = -1$

Solving Quadratic Equations Using the Square Root Property

Use the square root property when the equation can be written as $x^2 = a$.

• Example: $x^2 = 9$ $x = \pm 3$

Solving Quadratic Equations Using the Quadratic Formula

The quadratic formula is used when factoring is not possible.

• Example: $x^2 - 5x + 6 = 0$ $x = \frac{5 \pm \sqrt{25 - 24}}{2}$ $x = \frac{5 \pm 1}{2}$ $x = 3$ or $x = 2$

Linear Inequalities

Solving and Graphing Linear Inequalities

Linear inequalities are expressions involving <, >, ≤, ≥. Solutions are represented on a number line or using interval notation.

• Key Point 1: Solve inequalities using similar steps as equations, but reverse the inequality sign when multiplying/dividing by a negative number.

• Key Point 2: Graph solutions on a number line and use interval notation.

• Example: $x > 2$ is represented as $(2, \infty)$

Inequality	Solution Graph	Interval Notation
x < a	Arrow left from a	(-∞, a)
x > a	Arrow right from a	(a, ∞)
x ≤ a	Arrow left, closed at a	(-∞, a]
x ≥ a	Arrow right, closed at a	[a, ∞)

Inequality Properties

Properties of inequalities help in manipulating and solving them.

• Addition Property: If $a < b$, then $a + c < b + c$

• Subtraction Property: If $a < b$, then $a - c < b - c$

• Multiplication Property: If $a < b$ and $c > 0$, then $ac < bc$

• Division Property: If $a < b$ and $c > 0$, then $\frac{a}{c} < \frac{b}{c}$

Additional info:

Some examples and applications are inferred from context and standard college algebra curriculum, including the use of graphing calculators for checking solutions and interpreting word problems.