College Algebra: Equations, Inequalities, and Complex Numbers Study Guide

College Algebra: Equations, Inequalities, and Complex Numbers Study Guide

Equations and Inequalities

Solving Linear Equations

Linear equations are equations of the first degree, meaning the variable is raised only to the power of one. Solving these equations involves isolating the variable on one side of the equation.

• Key Point 1: To solve a linear equation, use inverse operations to isolate the variable.

• Key Point 2: Always simplify both sides of the equation as much as possible before solving.

• Example: Solve $x + 5 = 12$ Subtract 5 from both sides: $x = 7$

Solving Equations with Fractions

Equations may contain fractions. To simplify, multiply both sides by the least common denominator (LCD) to clear fractions.

• Key Point 1: Multiply both sides by the LCD to eliminate denominators.

• Key Point 2: After clearing fractions, solve as a standard linear equation.

• Example: Solve $\frac{x}{4} = 3$ Multiply both sides by 4: $x = 12$

Solving Equations with Variables on Both Sides

When variables appear on both sides, collect like terms and isolate the variable.

• Key Point 1: Move all terms containing the variable to one side and constants to the other.

• Key Point 2: Combine like terms before isolating the variable.

• Example: Solve $2x + 3 = x - 5$ Subtract $x$ from both sides: $x + 3 = -5$ Subtract 3: $x = -8$

Conditional, Identity, and Contradiction Equations

Equations can be classified based on their solution sets:

• Conditional Equation: True for some values of the variable.

• Identity: True for all values of the variable (e.g., $x = x$).

• Contradiction: False for all values of the variable (e.g., $0 = 1$).

Complex Numbers and Their Properties

Definition and Forms

A complex number is a number of the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, defined by $i^2 = -1$.

• Key Point 1: Real part: $a$; Imaginary part: $b$

• Key Point 2: Pure imaginary numbers have $a = 0$; real numbers have $b = 0$.

• Example: $3 + 4i$ is a complex number; $5i$ is pure imaginary; $7$ is real.

Operations with Complex Numbers

• Addition/Subtraction: Combine like terms: $(a + bi) + (c + di) = (a + c) + (b + d)i$

• Multiplication: Use distributive property and $i^2 = -1$: $(a + bi)(c + di) = (ac - bd) + (ad + bc)i$

• Example: $(2 + 3i) + (1 - 2i) = 3 + i$

• Example: $(2 + 3i)(1 - 2i) = 2 - 4i + 3i - 6i^2 = 2 - i + 6 = 8 - i$

Classifying Numbers

Numbers can be classified as real, imaginary, or complex. Some numbers are also rational or irrational.

• Real Numbers: All numbers on the number line, including rational and irrational numbers.

• Imaginary Numbers: Multiples of $i$ (e.g., $5i$).

• Complex Numbers: Numbers of the form $a + bi$.

• Example: $\sqrt{-9} = 3i$ is pure imaginary.

Quadratic Equations

Standard Form and Methods of Solution

A quadratic equation is an equation of the form $ax^2 + bx + c = 0$.

• Factoring: Express as a product of binomials and set each factor to zero.

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

• Completing the Square: Rewrite in the form $(x + p)^2 = q$ and solve for $x$.

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

• Example: Solve $x^2 - 4 = 0$ $x^2 = 4 \Rightarrow x = \pm2$

Discriminant and Nature of Solutions

The discriminant $D = b^2 - 4ac$ determines the nature of the solutions:

• If $D > 0$: Two distinct real solutions.

• If $D = 0$: One real solution (a repeated root).

• If $D < 0$: Two complex conjugate solutions.

• Example: For $x^2 + 4x + 5 = 0$, $D = 16 - 20 = -4$ (two complex solutions).

Inequalities

Solving Linear Inequalities

Linear inequalities are solved similarly to equations, but the direction of the inequality reverses when multiplying or dividing by a negative number.

• Key Point 1: Isolate the variable as with equations.

• Key Point 2: Reverse the inequality sign when multiplying/dividing by a negative.

• Example: Solve $-2x > 6$ Divide by $-2$ (reverse sign): $x < -3$

Interval Notation

Solutions to inequalities are often expressed in interval notation.

• Open Interval: $(a, b)$ means $a < x < b$

• Closed Interval: $[a, b]$ means $a \leq x \leq b$

• Example: $x > 2$ is $(2, \infty)$

Compound Inequalities

Compound inequalities involve two inequalities joined by "and" or "or".

• "And": Intersection of solution sets (e.g., $1 < x < 5$)

• "Or": Union of solution sets (e.g., $x < 1$ or $x > 5$)

Properties and Operations with Radicals

Simplifying Radicals

Radicals can be simplified by factoring out perfect squares and using the property $\sqrt{a^2} = a$.

• Key Point 1: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$

• Key Point 2: Rationalize denominators when necessary.

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

Operations with Radicals

• Addition/Subtraction: Combine like radicals (same radicand).

• Multiplication: Multiply radicands together: $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$

• Division: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

Tables

Classification of Numbers

Number	Real	Imaginary	Complex	Rational	Irrational
5	✔		✔	✔
$3i$		✔	✔
$\sqrt{2}$	✔		✔		✔
$2 + 3i$			✔

Additional info: Table entries inferred for clarity based on standard number classifications.

Summary

• Linear and quadratic equations can be solved using algebraic methods such as factoring, the quadratic formula, and completing the square.

• The discriminant determines the nature of solutions for quadratic equations.

• Complex numbers extend the real number system and have unique properties and operations.

• Inequalities are solved similarly to equations, with special attention to the direction of the inequality sign.

• Radicals and their operations are essential for simplifying expressions and solving equations involving roots.