Chapter 1: Equations, Inequalities, and the Rectangular Coordinate System

The Construction of the System: The rectangular coordinate system is established by drawing a horizontal line and a vertical line that intersect one another at right angles ( $90$ degrees).
The Axes:
- The x-axis: This is the horizontal line in the coordinate system.
- The y-axis: This is the vertical line in the coordinate system.
The Origin: The point where the x-axis and the y-axis intersect is called the origin. This point represents the zero points for both axes.
Numerical Orientation:
- Positive Numbers: On the x-axis, positive numbers are displayed to the right of the origin. On the y-axis, positive numbers are located above the origin.
- Negative Numbers: On the x-axis, negative numbers are displayed to the left of the origin. On the y-axis, negative numbers are located below the origin.

Ordered Pairs: Every point within the rectangular coordinate system corresponds to a specific ordered pair of real numbers, denoted as $(x, y)$ .
The x-coordinate: The first number in an ordered pair is the x-coordinate. It indicates the horizontal distance and the direction (left or right) from the origin along the x-axis.
The y-coordinate: The second number in an ordered pair is the y-coordinate. It indicates the vertical distance and the direction (up or down) from the origin along the y-axis.
Process for Plotting Specific Points:
- To Plot the Point $(-2, 4)$ : Start at the origin, move $2$ units to the left (negative direction on the x-axis), and then move $4$ units up (positive direction on the y-axis).
- To Plot the Point $(4, -2)$ : Start at the origin, move $4$ units to the right (positive direction on the x-axis), and then move $2$ units down (negative direction on the y-axis).

Defining Relationships: A relationship between two distinct quantities can be expressed mathematically as an equation involving two variables (typically $x$ and $y$ ). An example of such an equation is $y = x^2 - 4$ .
Solutions of Equations: A solution for an equation in two variables is an ordered pair of real numbers with the property that when the x-coordinate is substituted for $x$ and the y-coordinate is substituted for $y$ , the resulting equation is a true statement.

Procedure: To graph an equation like $y = |x + 1|$ using the point-plotting method, follow these steps:
1. Select x-values: Choose a range of integers for $x$ . In this specific instance, select integers starting with $-4$ and ending with $2$ .
2. Calculate y-values: For every chosen value of $x$ , determine the matching value for $y$ by substituting $x$ into the equation:
  - If $x = -4$ , then $y = |-4 + 1| = |-3| = 3$ , resulting in the point $(-4, 3)$ .
  - If $x = -3$ , then $y = |-3 + 1| = |-2| = 2$ , resulting in the point $(-3, 2)$ .
  - If $x = -2$ , then $y = |-2 + 1| = |-1| = 1$ , resulting in the point $(-2, 1)$ .
  - If $x = -1$ , then $y = |-1 + 1| = |0| = 0$ , resulting in the point $(-1, 0)$ .
  - If $x = 0$ , then $y = |0 + 1| = |1| = 1$ , resulting in the point $(0, 1)$ .
  - If $x = 1$ , then $y = |1 + 1| = |2| = 2$ , resulting in the point $(1, 2)$ .
  - If $x = 2$ , then $y = |2 + 1| = |3| = 3$ , resulting in the point $(2, 3)$ .
3. Plot and Connect: Plot these calculated points on the rectangular coordinate system and connect them to form the graph.

Graphing Utilities: Tools such as graphing calculators and computer software packages are known as graphers or graphing utilities.
The Viewing Rectangle: This is the portion of the coordinate plane displayed by the utility. It is defined by setting the minimum and maximum values for both the x-axis and the y-axis.
Standard Viewing Rectangle: The standard setting for most graphing utilities is denoted as $[-10, 10, 1]$ by $[-10, 10, 1]$ . This means the x-axis and y-axis both range from $-10$ to $10$ with a scale (distance between tick marks) of $1$ .
Interpreting Rectangle Notation: Symbols in the format $[Xmin, Xmax, Xscl]$ by $[Ymin, Ymax, Yscl]$ define the window properties:
- Example: A viewing rectangle of $[-100, 100, 50]$ by $[-100, 100, 10]$ signifies:
  - Minimum x-value: $-100$
  - Maximum x-value: $100$
  - Distance between x-axis tick marks: $50$
  - Minimum y-value: $-100$
  - Maximum y-value: $100$
  - Distance between y-axis tick marks: $10$

x-intercept: This is the x-coordinate of the point where a graph intersects the x-axis. At any x-intercept, the corresponding y-coordinate is always zero ( $0$ ).
- Example: If a graph crosses the x-axis at the point $(-3, 0)$ , the x-intercept is $-3$ .
y-intercept: This is the y-coordinate of the point where a graph intersects the y-axis. At any y-intercept, the corresponding x-coordinate is always zero ( $0$ ).
- Example: If a graph crosses the y-axis at the point $(0, 5)$ , the y-intercept is $5$ .

Application to Divorce Rates: Marriage longevity can be modeled using algebraic equations to examine social trends. For individuals married with a high school education but no college, the divorce rate can be modeled by the equation:
- $d = 1.8n + 14$
- In this formula, $d$ represents the percentage of marriages that end in divorce.
- The variable $n$ represents the number of years since the marriage began.
Predictive Calculation: To find the percentage of marriages ending in divorce after $15$ years for this demographic:
1. Substitute $n = 15$ into the equation: $d = 1.8(15) + 14$ .
2. Multiply $1.8 \times 15 = 27$ .
3. Add the constant: $27 + 14 = 41$ .
4. The result indicates that $41 \%$ of these marriages end in divorce after $15$ years.

The rectangular coordinate system is made up of a horizontal line (x-axis) and a vertical line (y-axis) that intersect at the origin.

The origin is the point where both axes meet, representing zero for both.
Positive numbers on the x-axis are to the right of the origin and on the y-axis above it. Negative numbers are to the left on the x-axis and below on the y-axis.

An ordered pair (x, y) represents a point in this system:
- The x-coordinate shows horizontal position from the origin.
- The y-coordinate shows vertical position from the origin.
Example: To plot (-2, 4), move 2 units left and 4 units up from the origin.

An equation involving two variables (like y = x² - 4) shows a relationship between x and y values.

Divorce rates can be modeled with equations to understand trends, like d = 1.8n + 14 for marriages after n years.