College Algebra Study Guide: Equations and Inequalities

Ch. 1 – Equations and Inequalities

Solving Linear Equations

Linear equations are equations where the variable appears to the first power and is not multiplied by itself. Solving these equations is a foundational skill in algebra.

• Linear Expression: An algebraic expression of the form ax + b.

• Linear Equation: An equation that can be written as ax + b = c.

• Unknown: The variable to solve for.

• Operations: Use addition, subtraction, multiplication, or division to isolate the variable.

Example: Solve $2x + 3 = 11$

• Subtract 3 from both sides: $2x = 8$

• Divide both sides by 2: $x = 4$

Key Steps for Solving Linear Equations:

1. Distribute where necessary

2. Combine like terms

3. Isolate the variable using inverse operations

4. Check the solution by substituting into the original equation

Linear Equations with Fractions

Some linear equations contain fractions. To solve these, clear fractions by multiplying both sides by the least common denominator (LCD).

Example: Solve $\frac{1}{2}(x + 2) = \frac{3}{4}$

• Multiply both sides by 4 (LCD): $2(x + 2) = 3$

• Expand: $2x + 4 = 3$

• Solve: $2x = -1 \implies x = -\frac{1}{2}$

Categorizing Linear Equations

Linear equations can be classified based on the number of solutions:

• Identity: True for all real numbers (infinite solutions)

• Conditional: True for one value (one solution)

• Inconsistent: Not true for any value (no solution)

Example Table:

Equation	Type	Solution Set
$2x + 4 = 10$	Conditional	$x = 3$
$2x + 5 = 2x + 5$	Identity	All real numbers
$2x + 5 = 2x + 3$	Inconsistent	No solution

Powers of i

The imaginary unit i is defined as $i = \sqrt{-1}$. Powers of i repeat in a cycle of four:

• $i^1 = i$

• $i^2 = -1$

• $i^3 = -i$

• $i^4 = 1$

Any higher power of i can be simplified using this cycle.

Example: $i^7 = i^{4 \times 1 + 3} = (i^4)^1 \cdot i^3 = 1 \cdot (-i) = -i$

Complex Numbers

Introduction to Complex Numbers

A complex number is a number of the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part.

• Standard Form: $a + bi$

• Real Part: $a$

• Imaginary Part: $b$

Example: $3 + 2i$ (real part: 3, imaginary part: 2)

Adding and Subtracting Complex Numbers

Add or subtract the real parts and the imaginary parts separately.

Example: $(4 + 6i) + (3 - 2i) = (4 + 3) + (6i - 2i) = 7 + 4i$

Multiplying Complex Numbers

Multiply as you would binomials (FOIL), using $i^2 = -1$ to simplify.

Example: $(3 + 2i)(4 - i) = 12 - 3i + 8i - 2i^2 = 12 + 5i + 2 = 14 + 5i$

Complex Conjugates

The conjugate of $a + bi$ is $a - bi$. Multiplying a complex number by its conjugate yields a real number:

• $(a + bi)(a - bi) = a^2 + b^2$

Dividing Complex Numbers

To divide by a complex number, multiply numerator and denominator by the conjugate of the denominator.

Example: $\frac{3}{1 + 2i} \times \frac{1 - 2i}{1 - 2i} = \frac{3(1 - 2i)}{(1 + 2i)(1 - 2i)} = \frac{3 - 6i}{1 - 4i^2} = \frac{3 - 6i}{1 + 4} = \frac{3 - 6i}{5} = \frac{3}{5} - \frac{6}{5}i$

Intro to Quadratic Equations

Factoring Quadratics

A quadratic equation is an equation of the form $ax^2 + bx + c = 0$. Factoring is one method to solve quadratics.

Factoring Steps:

1. Write in standard form

2. Factor completely

3. Set each factor equal to zero

4. Solve for $x$

The Square Root Property

For equations of the form $x^2 = k$, take the square root of both sides:

• $x = \pm \sqrt{k}$

Example: $x^2 = 9 \implies x = \pm 3$

The Quadratic Formula

The quadratic formula solves any quadratic equation $ax^2 + bx + c = 0$:

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

The discriminant $b^2 - 4ac$ determines the number and type of solutions:

Discriminant	Number of Solutions	Type
Positive	2	Real
Zero	1	Real (double root)
Negative	2	Complex

Completing the Square

To solve $ax^2 + bx + c = 0$ by completing the square:

1. Move $c$ to the other side

2. Divide by $a$ if necessary

3. Add $(\frac{b}{2a})^2$ to both sides

4. Write as a perfect square and solve

Solving Rational Equations

A rational equation contains fractions with variables in the denominator. To solve:

1. Find the least common denominator (LCD)

2. Multiply both sides by the LCD to clear denominators

3. Solve the resulting equation

4. Check for extraneous solutions (values that make any denominator zero)

Linear Inequalities

Interval Notation

Interval notation is a concise way to describe solution sets:

• Closed Interval: $[a, b]$ includes endpoints

• Open Interval: $(a, b)$ excludes endpoints

• Half-Closed Interval: $[a, b)$ or $(a, b]$ includes one endpoint

Solving Linear Inequalities

Solve inequalities as you would equations, but reverse the inequality sign when multiplying or dividing by a negative number.

Example: $-2x > 6 \implies x < -3$

Fractions & Variables on Both Sides

When solving inequalities with fractions or variables on both sides, clear denominators and collect like terms as with equations.

Additional info: This guide covers all major topics from Ch. 1 – Equations and Inequalities, including linear and quadratic equations, complex numbers, rational equations, and inequalities, with definitions, examples, and step-by-step methods.