Study Notes: Equations and Inequalities (Precalculus, Ch. 1)

Rectangular Coordinate System

Graphing in the Rectangular Coordinate System

The rectangular (Cartesian) coordinate system is used to graph points, lines, and equations in two dimensions. It consists of two perpendicular number lines (axes) that intersect at the origin (0,0), dividing the plane into four quadrants.

• Ordered pairs (x, y) specify the position of points.

• x-axis: horizontal axis; y-axis: vertical axis.

• Quadrants are numbered I to IV, counterclockwise from the upper right.

Example: Plot the points A(4, 3), B(-2, 2), C(-2, -3), D(-4, 0), E(0, 5) on the graph. Identify their quadrants.

Equations of Two Variables

Solving Two-Variable Equations

Equations can involve one or two variables. Two-variable equations are graphed as lines or curves in the coordinate plane.

• Equation with one variable: Solution is a point on a number line.

• Equation with two variables: Solution is a set of points (x, y) on a plane.

Example: Graph the equation -2x + y = -1 by creating ordered pairs for x = -2, -1, 0, 1, 2.

Graphing by Plotting Points

• Isolate y (or x) in the equation.

• Choose 3–5 values for x (or y), calculate corresponding y (or x) values.

• Plot the points and connect with a straight line or smooth curve.

Practice: Graph y - x^3 + 3 = 9 and y = (x^2) + 1 by plotting points.

Intercepts

Graphing Intercepts

Intercepts are points where a graph crosses the x-axis or y-axis.

• x-intercept: Set y = 0, solve for x.

• y-intercept: Set x = 0, solve for y.

Example: Find the intercepts of y = 2x - 4 and x^2 + y^2 = 9.

Solving Linear Equations

Linear Equations and Expressions

A linear equation is an equation of the form ax + b = 0. To solve, isolate the variable using inverse operations.

• Simplify both sides if needed.

• Apply the same operation to both sides to maintain equality.

Example: Solve 2(x - 3) = 0.

Linear Equations with Fractions

• Multiply both sides by the least common denominator (LCD) to clear fractions.

• Solve as usual.

Example: Solve (x + 2)/3 = 1/2.

Categorizing Linear Equations

Linear equations can have:

• One solution (conditional)

• No solution (inconsistent)

• Infinitely many solutions (identity)

Example: Solve and categorize 2x + 10 = 10, 2x + 5 = 2x + 3, and 2x + 5 = 2x + 5.

Solving Rational Equations

Rational Equations

A rational equation contains one or more rational expressions (fractions with variables in the denominator). To solve:

• Multiply both sides by the LCD to clear denominators.

• Solve the resulting equation.

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

Example: Solve 1/(x - 1) = 12.

Complex Numbers

Introduction to Complex Numbers

A complex number is of the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part ($i = \sqrt{-1}$).

• Combine like terms when adding or subtracting complex numbers.

Example: Add (4 + 2i) + (6 - i).

Multiplying Complex Numbers

• Use the distributive property (FOIL) and simplify using $i^2 = -1$.

Example: (3 + 2i)(4 - 2i).

Complex Conjugates

• The conjugate of $a + bi$ is $a - bi$.

• Multiplying a complex number by its conjugate yields a real number: $(a + bi)(a - bi) = a^2 + b^2$.

Example: Find the product of (2 + 3i)(2 - 3i).

Dividing Complex Numbers

• Multiply numerator and denominator by the conjugate of the denominator to rationalize.

Example: $\frac{3}{1 + 2i}$

Intro to Quadratic Equations

Factoring Quadratic Equations

A quadratic equation is of the form $ax^2 + bx + c = 0$. To solve by factoring:

• Set the equation to zero.

• Factor the quadratic expression.

• Set each factor to zero and solve for x.

Example: $x^2 + 12x = 0$

The Square Root Property

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

• Apply to equations in the form $(x + a)^2 = k$.

Example: $(x + 3)^2 = 9$

Completing the Square

• Rewrite $ax^2 + bx + c = 0$ in the form $(x + d)^2 = k$ by adding and subtracting the same value.

• Solve using the square root property.

Example: $x^2 + 6x + 5 = 0$

The Quadratic Formula

• The solutions to $ax^2 + bx + c = 0$ are given by:

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

Example: Solve $3x^2 + 4x + 1 = 0$ using the quadratic formula.

The Discriminant

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

Discriminant	Number/Type of Solutions
Positive	2 real solutions
Zero	1 real solution
Negative	2 complex solutions

Linear Inequalities

Interval Notation

Interval notation is a concise way to express solution sets for inequalities.

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

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

• Half-closed interval: $[a, b)$ or $(a, b]$ includes one endpoint.

Example: Express $1 x 5$ in interval notation: [1, 5]

Solving Linear Inequalities

• Solve as you would a linear equation, but reverse the inequality sign when multiplying or dividing by a negative number.

Example: Solve $-2x + 5 < 6$ and express the solution in interval notation.

Fractions & Variables on Both Sides

• Clear fractions by multiplying both sides by the LCD.

• Isolate the variable and solve.

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

Additional info: These notes cover the foundational concepts of equations and inequalities, including graphing, solving, and classifying equations, as well as working with complex numbers and quadratic equations. They are suitable for Precalculus students preparing for exams or reviewing key algebraic methods.