Equations and Inequalities: Foundations for Precalculus

Rectangular Coordinate System

Introduction to the Cartesian Plane

The rectangular coordinate system, also known as the Cartesian plane, is a fundamental tool for graphing equations and visualizing relationships between variables in algebra and precalculus. It consists of two perpendicular axes that form a two-dimensional plane.

• Horizontal axis: The x-axis

• Vertical axis: The y-axis

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

• Origin: The point (0, 0) where the x and y axes intersect

• Quadrants: The axes divide the plane into four quadrants, numbered counter-clockwise starting from the top-right

• Sign conventions: x values are positive to the right of the origin, negative to the left; y values are positive above the origin, negative below

Plotting Points and Identifying Quadrants

To plot points, locate the x value on the horizontal axis and the y value on the vertical axis. The quadrant of a point depends on the signs of its coordinates:

• Quadrant I: x > 0, y > 0

• Quadrant II: x < 0, y > 0

• Quadrant III: x < 0, y < 0

• Quadrant IV: x > 0, y < 0

Equations of Two Variables

Solving and Graphing Two-Variable Equations

Many equations in algebra involve two variables, typically x and y. The graph of such an equation represents all (x, y) pairs that satisfy the equation.

• To check if a point satisfies an equation, substitute its x and y values into the equation.

• If the equation is true, the point lies on the graph; otherwise, it does not.

Graphing by Plotting Points

To graph an equation with two variables:

1. Isolate y (if possible): y = ...

2. Choose 3-5 x-values and calculate corresponding y-values

3. Plot the resulting (x, y) points

4. Connect the points with a line or curve

Intercepts

Finding x- and y-Intercepts

Intercepts are points where a graph crosses the axes:

• x-intercept: The value of x when y = 0

• y-intercept: The value of y when x = 0

• Intercepts are written as ordered pairs

Solving Linear Equations

Steps for Solving Linear Equations

A linear equation is an equation involving a variable to the first power. 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

When fractions are present, multiply both sides by the least common denominator (LCD) to clear fractions before proceeding.

Categorizing Linear Equations

Linear equations can be classified based on their solution sets:

• Conditional: One solution

• Identity: All real numbers are solutions

• Inconsistent: No solution

Solving Rational Equations

Definition and Solution Process

A rational equation contains a variable in the denominator. To solve:

1. Identify restrictions by setting denominators equal to zero

2. Multiply both sides by the LCD to eliminate fractions

3. Solve the resulting linear equation

4. Check solutions against restrictions

Complex Numbers

Introduction and Standard Form

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

• Real part: a

• Imaginary part: b

Operations with Complex Numbers

• Add/Subtract: Combine like terms, express in standard form

• Multiply: Use distributive property or FOIL, simplify $i^2 = -1$

• Conjugate: Change the sign of the imaginary part; product of a complex number and its conjugate is always real

• Divide: Multiply numerator and denominator by the conjugate of the denominator to obtain a real denominator

Quadratic Equations

Standard Form and Factoring

A quadratic equation is a polynomial of degree 2, written as $a x^2 + b x + c = 0$. To solve by factoring:

1. Write in standard form

2. Factor completely

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

4. Check solutions

Square Root Property

For equations of the form $(x + h)^2 = k$, solve by taking the square root of both sides:

• Isolate the squared expression

• Take both positive and negative square roots

• Solve for x

Completing the Square

If a quadratic is not easily factorable, rewrite it in the form $(x + h)^2 = k$ by completing the square:

1. Simplify to $x^2 + bx = c$

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

3. Factor left side as a perfect square

4. Solve using the square root property

Quadratic Formula

The quadratic formula solves any quadratic equation:

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

• Plug in values for a, b, and c from the standard form

The Discriminant

The discriminant ($b^2 - 4 a c$) determines the number and type of solutions:

• Positive: Two real solutions

• Zero: One real solution

• Negative: Two complex solutions

Choosing a Method to Solve Quadratic Equations

Depending on the form of the equation, choose the most efficient method: factoring, square root property, completing the square, or quadratic formula.

Linear Inequalities

Interval Notation and Set Builder Notation

Solution sets for inequalities can be expressed in interval notation, set builder notation, or graphed on a number line.

• Closed interval: [a, b] means a ≤ x ≤ b

• Open interval: (a, b) means a < x < b

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

• Use infinity symbols for unbounded intervals: (−∞, a), (b, ∞)

Solving Linear Inequalities

Linear inequalities are solved similarly to linear equations, but remember:

• Multiplying or dividing both sides by a negative number reverses the inequality symbol

• Express solution sets in interval notation

Fractions and Variables on Both Sides

Clear fractions by multiplying by the LCD, then solve as usual. Express the solution set in interval notation and graph.