A Linear equation in two variables has 3 main forms:

(1) Standard form: ax + by = c or ax + by + c = 0

(2) Slope-intercept form: y = mx + b, where m is the slope and b is the y-coordinate of the y-intercept (0, b).

(3) Point-slope form: y = m(x – x₁) + y₁, where m is the slope and (x₁, y₁) is a point on the line. Another version of this form is: y – y₁ = m(x – x₁)

Inequality: a mathematical statement which indicates two unequal numbers. The verb is usually translated to one of these symbols: <, ≤, >, ≥

critical point: the point on the number line that separates the solution interval from the rest of the number line. May (closed point) or may not (open point) be part of the solution interval.

closed: a critical point is closed when it IS part of the solution interval. We recognize a closed point by the inequality signs ≥ and ≤. In Interval Notation, closed points get brackets [-a, b].

open: a critical point is open when it IS NOTpart of the solution interval. We recognize an open point by the inequality signs > and <. A positive or negative infinity sign is ALWAYS open. In Interval Notation, open points get parentheses (-a, b).

Three ways to indicate the solution set of an inequality, for example 3 < x < 5

1. Graphical solution on a number line
2. Interval Notation to show the beginning and ending values (3, 5)
3. Set Builder notation {x| x is a Real number and 3 < x < 5 }

Two lines are parallel when their graphs do not intersect. When the equations are in Slope-Intercept form, the slopes are the same but the y-intercepts are different.

Two lines are perpendicular when their graphs intersect at right angles (90 degrees). When the equations are in Slope-Intercept form, the slopes are Negative–Reciprocals, that is, the slopes have opposite signs and are flipped. Example. 2/3 and -3/2

The Slope gives the directions from one point on a line to another point on that same line. Given as Rise over Run ( Rise/Run ), where the Rise is the difference in y-coordinates, and Run is the difference in x-coordinates of the two points.

Union: The set formed from by joining the elements of two sets. An element of a set is put into the union if it is in the first set OR the second set. Keyword is OR.
Example. If the set A = {1, 3, 5, 7, 9} and the set B = { 3, 4, 5, 6, 7, 8}, then AUB = { 1, 3, 4, 5, 6, 7, 8, 9 }

Intersection. The set formed from the common elements of two sets. An element of a set is put into the intersection if it is in the first set AND the second set (must be in both sets at the same time). Keyword is AND.
Example. If the set A = {1, 3, 5, 7, 9} and the set B = { 3, 4, 5, 6, 7, 8}, then A∩B = { 3, 5, 7 }

Model: an abstraction that has 2 characteristics: (1) It can explain present phenomena (events, conditions, results) and should NOT contradict data and information known to be correct. (2) It should be able to make predictions about data or results, using current information to forecast phenomena or create new information.

Mathematical models are used to forecast business trends, design the shape of cars, estimate ecological events (weather forecast maps and storm predictors that TV weather people use), and discover new information when human knowledge is inadequate.

Interpolation: estimating values between data points. More accurate.

Extrapolation: estimating (predicting) values beyond the given data points.

X-intercept: the point (a, 0) where the graph intersects (crosses) the x-axis.

Zero of the function: the x-coordinate of the x-intercept. It is the input that returns an output of 0.

Piecewise–defined function: when a function models real world data, one formula for f(x) may not work, so the function is defined along pieces of its domain. An example is the Fijita Scale for classifying the intensity of tornados based on wind speed. Since the graph of the Fijita Scale looks like steps, it is also called a "step-function." Another example is the grading scale for this class.

Continuous: a piece-wise function is continuous if you can draw the graph without picking up your pencil. If one or more breaks occur in the graph, such as with a step function, then the function is discontinuous.

The absolute value of a number is its distance from the origin (0) on the number line, regardless of its direction. The symbol | #| is used to indicate the absolute value of #. For example, |4| = 4 and |–4| = 4, since both 4 and –4 are four units from 0.

Absolute value equation: an equation with a variable inside an absolute value symbol; for example, |x – 4| = 9. To solve an abosolute value equation, we MUST investigate both possiblilties: (1) The expression x – 4 represents a –9 OR (2) the expression x – 4 represents a +9 (I call these two-fors, because you get two problems for one). The smallest an absolute value can be is 0.

Absolute value inequality: an inequality with a variable inside an absolute value symbol; for example, |x – 4| < 9 or |x| > 7. To solve an absolute value inequality, we investigate both possiblilties as though the inquality is an equation to find the critical points, then determine whether the solution interval is continuous, an intersection, or discontinuous, a union. 1. For absolute value inequalities of the type |x| < a, where a is any real number > 0, then the solution interval is an intersection between –a and +a. The interval looks like (–a, a). 2. For absolute value inequalities of the type |x| > a, where a is any real number > 0, then the solution interval is a union of the intervals (–∞, –a) (+a,∞) .