College Algebra: Equations, Inequalities, and Complex Numbers

Ch. 1 - Equations and Inequalities

Rectangular Coordinate System

The rectangular coordinate system, also known as the Cartesian plane, is a two-dimensional plane formed by two perpendicular axes: the horizontal x-axis and the vertical y-axis. This system is fundamental for graphing equations and visualizing relationships between variables.

• Axes: The x-axis is horizontal; the y-axis is vertical.

• Ordered pairs: Points are written as (x, y).

• Origin: The point (0, 0) where the axes intersect.

• Signs: x-values are positive to the right, negative to the left; y-values are positive above, negative below the origin.

• Quadrants: The axes divide the plane into four quadrants, numbered counterclockwise starting from the upper right.

Example: Plot points such as A(4, 3), B(−3, 2), C(−2, −3), D(5, −4), E(0, 0), F(0, −3) and identify their quadrants.

Solving Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The general form is ax + b = c.

• Operations: Use addition, subtraction, multiplication, and division to isolate the variable x. Always perform the same operation on both sides of the equation.

• Steps to Solve:

  1. Distribute constants

  2. Combine like terms

  3. Group terms with x and constants on opposite sides

  4. Isolate x

  5. Check the solution by substituting x back into the original equation

• Linear equations with fractions: Multiply both sides by the least common denominator (LCD) to clear fractions before solving.

Example: Solve 2(x−3) = 0 by distributing, isolating x, and checking the solution.

Categorizing Linear Equations

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

• Conditional Equation: Has exactly one solution (e.g., 2x + 4 = 10).

• Identity: True for all real numbers (e.g., x + 5 = x + 5).

• Inconsistent Equation: Has no solution (e.g., x = x + 4).

Example: Solve and categorize equations such as 5x + 17 = 8x + 12 − 3(x + 4).

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. The imaginary unit i is defined as $i = \sqrt{-1}$.

• Standard form: a + bi

• Examples: 2 + 0i (purely real), 0 + 7i (purely imaginary), 4 − 3i (complex)

Practice: Identify the real and imaginary parts of given complex numbers.

Adding & Subtracting Complex Numbers

To add or subtract complex numbers, combine like terms (real with real, imaginary with imaginary). Always express the answer in standard form.

• Example: (2 + 8i) − (4 − i) = (2 − 4) + (8i + i) = −2 + 9i

Multiplying Complex Numbers

Multiply complex numbers as you would binomials (using the distributive property or FOIL), then simplify using $i^2 = -1$.

• Steps:

  1. Distribute or FOIL

  2. Apply $i^2 = -1$

  3. Combine like terms

• Example: (3 + 8i)^2

Complex Conjugates

The conjugate of a complex number a + bi is a − bi. Multiplying a complex number by its conjugate always results in a real number.

• Formula: $(a + bi)(a - bi) = a^2 + b^2$

• Example: (2 + 3i)(2 − 3i) = 4 + 9 = 13

Dividing Complex Numbers

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

• Steps:

  1. Multiply numerator and denominator by the conjugate of the denominator

  2. Expand and simplify

  3. Express the result in standard form

• Example: $\frac{6 + i}{4 - 2i}$

Powers of i

The powers of i repeat in a cycle of four:

• $i^1 = i$

• $i^2 = -1$

• $i^3 = -i$

• $i^4 = 1$

To simplify higher powers, divide the exponent by 4 and use the remainder to determine the result.

• Example: $i^{22} = (i^4)^5 \cdot i^2 = 1^5 \cdot (-1) = -1$

Quadratic Equations

Factoring Quadratic Equations

A quadratic equation is an equation of the form $ax^2 + bx + c = 0$. Factoring is one method to find its solutions (roots).

• Steps:

  1. Write the equation in standard form

  2. Factor completely

  3. Set each factor equal to zero and solve for x

  4. Check solutions in the original equation

• Example: $x^2 + x - 6 = 0$ factors to $(x + 3)(x - 2) = 0$ so $x = -3$ or $x = 2$

The Square Root Property

When a quadratic equation is in the form $(x + h)^2 = k$, solve by taking the square root of both sides.

• Steps:

  1. Isolate the squared expression

  2. Take the positive and negative square roots

  3. Solve for x

• Example: $(x - 1/2)^2 - 5 = 0$ leads to $x - 1/2 = \pm \sqrt{5}$

Imaginary roots occur when taking the square root of a negative number, resulting in complex solutions.

Completing the Square

If a quadratic is not easily factorable, you can rewrite it in the form $(x + p)^2 = q$ by completing the square.

• Steps:

  1. Move the constant term to the other side

  2. Add $(b/2)^2$ to both sides

  3. Factor the left side as a perfect square

  4. Solve using the square root property

• Example: $x^2 + 6x + 1 = 0$ becomes $(x + 3)^2 = 8$

The Quadratic Formula

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

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

• Plug in the values of a, b, and c, then compute and simplify the solutions.

• Example: $x^2 - 5x = -1$

The Discriminant

The discriminant is the expression under the square root in the quadratic formula: $b^2 - 4ac$. It determines the number and type of solutions:

• If $b^2 - 4ac > 0$: Two distinct real solutions

• If $b^2 - 4ac = 0$: One real solution (a repeated root)

• If $b^2 - 4ac < 0$: Two complex (imaginary) solutions

Solving Rational Equations

Rational Equations

A rational equation contains a variable in the denominator. To solve, clear denominators by multiplying both sides by the least common denominator (LCD), then solve the resulting equation.

• Restrictions: Exclude any value that makes a denominator zero.

• Steps:

  1. Find restrictions by setting denominators equal to zero

  2. Multiply both sides by the LCD

  3. Solve the resulting equation

  4. Check that solutions do not violate restrictions

Example: Solve $\frac{x}{x-1} = \frac{7}{6}$, $x \neq 1$

Linear Inequalities

Interval Notation

Interval notation is a concise way to describe sets of numbers, especially solution sets for inequalities.

• Closed interval: [a, b] includes endpoints a and b.

• Open interval: (a, b) excludes endpoints a and b.

• Half-open intervals: [a, b) or (a, b]

• Infinity: Use (−∞, a) or (a, ∞) with parentheses, since infinity is not a number.

Solving Linear Inequalities

Linear inequalities are solved similarly to linear equations, but with an inequality symbol (>, <, ≥, ≤) instead of =. When multiplying or dividing both sides by a negative number, reverse the inequality symbol.

• Example: Solve 2x + 12 > 19, express the solution in interval notation, and graph.

Inequalities with Fractions and Variables on Both Sides

Solve as you would a linear equation: clear fractions, collect like terms, and isolate the variable. Remember to reverse the inequality if multiplying or dividing by a negative.

• Example: $\frac{1}{3}(x + 1) \geq \frac{1}{5}(3 + 2x)$