Basic Equations and Inequalities

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 1 - of 45 Topic 0: Definition: Ex. 1 Basic Equations and Inequalities An equation is a statement that the values of two expressions are equal. Show that x = 5 is a solution to 3x 4 = 2x + 1. Show that x = 2 is not. Two primary rules can be used when solving a basic equation: 1. Simplify each expression on either side of the relationship independent of the other side. 2. An action which changes an expression on one side of a relationship must be matched by an equivalent change on the other side of the relationship. Ex. 3 Solve. 3x 2 14x

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 2 - of 45 In some environments, the solutions to an equation may be expressed as a set. Definition: A set is a well-defined collection of distinct objects. Sets are usually expressed using braces, also referred to as curly brackets. If x = 5 is the only solution to 3x 4 = 2x + 1, then we can refer to {5} as the solution set of the equation. Some basic equations can be solved by every real number. A short hand way of expressing the set of every real number is which is called blackboard r. Definition: An equation where every real number solves the equation is called an identity statement. Some basic equations cannot be solved by any real number. A set without any object in it is called an empty set and is expressed by. Definition: An equation where no real number exists to solve the equation is called a contradiction.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 3 - of 45 Definition: An inequality is a statement that the values of two expressions are not equal. Examples of inequality relationships include: Ex. 2 Determine whether 5 is a solution to each inequality below. a. 2x x + 2 a < b a > b a b a is (strictly) less than b a is (strictly) greater than b a is not equal to b Inequality relationships that include the possibility for equality as well include: a < b a > b a is less than or equal to b a is greater than or equal to b Note that the use of the word or implies two equally possible relationships. b. x 1 > 6x c. 4x 7<x +8

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 4 - of 45 The primary rules for solving basic equations can be applied to inequalities if one more rule is added: 3. When multiplying or dividing by a negative number, reverse the direction of the inequality symbol. Ex. 4 Solve. 352x 1 A compound inequality, also called a three-part inequality, is an inequality which relates three expressions using two (like-directed) inequality symbols. The simplest compound inequalities will involve a single variable expression between two numeric expressions can be solved using the same rules of basic inequalities. Ex. 6 Solve. 5 2x 113 Ex. 5 Solve. x 4 2

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 5 - of 45 Typically, inequalities have many solutions, a significant difference from the usually limited number of solutions an equation has. While a special form of a set can be created to express the solutions within a set, typically the solutions to inequalities are expressed using interval notation. Ex. 7 Express the solution inequalities from previous examples in interval notation. a. Ex. 4 Interval Notation ( ) parentheses boundary excluded or, or brackets [ ] boundary included b. Ex. 5 c. Ex. 6 left end boundary or right end boundary or

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 6 - of 45 Symbols/terminology relevant to intervals & interval notation: Ex. 8 Classify the intervals found in previous examples. Closed Interval: 2,7, 5, any and all numeric boundaries are included a. Ex. 4 Open Interval: 4,3,,4 any and all numeric boundaries are excluded Half-open Interval: 5, 1, 0,4 one end open, one end closed b. Ex. 5 Infinite Interval:,4, 5,, unbounded on one or both ends, Union Symbol: <means or> each entire interval appropriate c. Ex. 6 Intersection Symbol: <means and> only common (i.e. overlapping) interval appropriate

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 7 - of 45 Another way of expressing the solutions to an inequality is to sketch a graph of the solutions on a number line. We graph inequalities so that we can see the solutions. To graph the solution set of an inequality: Ex. 9 Sketch a graph of the solution set of each inequality from previous examples. a. Ex. 4 1. Identify the boundaries of the solution interval. 2. Use an appropriate symbol to express whether the boundary is included or excluded a. Use a closed dot if the boundary is included. b. Use an open dot if the boundary is excluded. 3. Shade appropriately, using test values as necessary to identify the intervals. b. Ex. 5 c. Ex. 6

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 8 - of 45 Topic 1: The Coordinate Plane; Symmetry Overview of properties and terminology associated with the coordinate plane: Definition: Symmetry is the property of having exact correspondence of form and configuration when an image is reflected about an axis or rotated about a point. Two primary forms of symmetry: Reflective Symmetry: Rotational Symmetry:

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 9 - of 45 On the coordinate plane: The point (c, b) is symmetric about the x-axis to the point (a, b) if c = a. or in other words, a point (x, y) can be reflected about the x-axis by replacing y with y. The point (a, d) is symmetric about the y-axis to the point (a, b) if d = b. or in other words, a point (x, y) can be reflected about the y-axis by replacing x with x. The point (c, d) is symmetric about the origin to the point (a, b) if c a and d = b. or in other words, a point (x, y) can be rotated about the x-axis by replacing both x with x and y with y. Ex. 1 Plot points on the coordinate plane that satisfy the condition of symmetry given. A B C a. Point D is symmetric about the x-axis to A. b. Point E is symmetric about the y-axis to B. c. Point F is symmetric about the origin to C.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 10 - of 45 Topic 2: Distance and Midpoint The distance d across the coordinate plane from point x, y to point x, y is determined by 1 1 2 2 2 2 2 1 2 1. d x x y y Ex. 1a Find the distance between 3, 2 1, 4. and The midpoint M of a line segment connecting point x1, y1 to point x2, y2 is determined by x1x2 y1y2 Mxy,,. 2 2 Ex. 1b Find the midpoint between 3, 2 1, 4. and

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 11 - of 45 Ex. 2 Find the distance from A to B and the location of the midpoint between A & B. A 2, 5 B1,4

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 12 - of 45 Topic 3: Definition: Linear Equations of Two Variables; Graphs of Equations of Two Variables A linear equation of two variables is an equation which can be expressed in the form Ax + By = C, where A, B, and C are real numbers and both A and B are not zero. Solutions to an equation of two variables are expressed as ordered pairs (x, y). The graph of a linear equation of two variables produces a line on the coordinate plane. Other important forms of linear equations include: Slope-intercept form: y = mx + b, m is the slope includes the point (0, b) Point-slope form: y y 1 = m(x x 1 ) m is the slope includes the point (x 1, y 1 ) Vertical line: Horizontal line: x = a, a is a real number y = b, b is a real number Linear equations are examples of relations. Definition: A relation is a set of points in the plane. Recall that the slope of a line is the value found by the ratio of vertical change to horizontal change of two points on the line. Horizontal lines have a slope of zero while vertical lines do not have slope. Note that relations may or may not be defined by equations at all. Slope formula: y y m x x 2 1 2 1

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 13 - of 45 Ex. 1 Find the slope-intercept form of the line Ex. 2 Restate the equation in slope-intercept form, identify the slope, and sketch a graph of the line on the coordinate plane. 3x + 4y = 12

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 14 - of 45 Although we will use strategies in the future for producing graphs that occur in common or important situations, it is worth noting that a sketch can be approximated by creating a table of values, assigning values for one variable and calculating the value of the other variable. Ex. 3 Create a table of values to sketch a graph of the equation. 2 y x 3x

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 15 - of 45 Topic 4: Definition: Linear Systems of Two Variables A system of equations is a set of equations involving the same variables simultaneously. A system of equations is said to be solved when a set of values for each variable in the system satisfies every equation in the system. Systems of linear equations are those systems where every equation is linear. In college algebra we will want to solve systems of linear equations. A system of nonlinear equations is a system with one or more nonlinear equation and will not be covered here. Solving a linear system of two variables can be accomplished in four ways: 1. Finding the intersection point of the graphs of each equation. 2. Substituting one equation into the other to find the value of one variable and then using that value to find the value of the second variable. 3. Adding equations together, after appropriate multiplicative operations to one or both equations, to find the value of one variable. Then using that value to find the value of the second variable. 4. Corresponding the coefficients of the system to a matrix and applying elementary row operations to create an identity matrix and a solution matrix. We will not apply matrices to solve systems in this class. Our primary approaches will be methods 2 and 3, the Substitution Method and the Elimination method, respectively, with supplemental graphing for confirmation.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 16 - of 45 The Substitution Method requires the following set of steps: Ex. Solve the system of equations using the Substitution Method. 1. Solve either equation for x or y. 2. Plug that equation in for the variable in the other equation. 3. Solve the created equation involving one variable. 4. Plug the solution found in step 3 into any equation involving two variables & solve. 5. Write your solution as an ordered pair. x 4y 4 3x 2y 16 The Addition/Elimination Method requires the following set of steps: 1. As necessary, multiply one or both coefficients by a positive or negative whole number so that a column of coefficients have the same absolute value but different signs. 2. Add the equations to eliminate a variable and create a new equation. 3. Plug the solution found in step 2 into any equation involving two variables & solve. 4. Write your solution as an ordered pair.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 17 - of 45 Ex. Solve the system of equations using the Elimination Method. 2x y 1 3x 2y 4 Observe that if each system is graphed, the solution corresponds with the intersection point of the equations in the system.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 18 - of 45 Ex. 3 Solve the system of equations. 5x 3y 2 2x 4y 6 Most systems of linear equations have exactly one solution. A system that has exactly one solution is classified as being consistent and independent. A system of linear equation can have no solution if the graphs of the equations are parallel (as thus lack an intersection point). Such a system is classified as being inconsistent. When solving a system through either method, if a contradiction is reached (such as two different numbers supposedly being equal to each other), the system must be inconsistent. A system of linear equations can have infinitelymany solutions if the equations are not unique (and thus share all their points along a coincident line). Such a system is classified as being dependent. When solving a system through either method, if an identity statement is reached (such as a number being equal to itself), the system must be dependent.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 19 - of 45 Topic 5: Linear Systems of Three Variables To solve a linear system of three variables, we will follow the following set of steps: 1. Write each equation in standard form Ax + By + Cz = D 2. Choose a pair of equations and eliminate one of the variables by the Addition/Elimination method. 3. Choose a different pair of equations and eliminate the same variable. 4. Utilize the equations in steps 2 and 3 to solve for one variable by either the Substitution or Addition/Elimination method. Then find the value of the other variable in those equations. 5. Plug the values found in step 4 into any of the original equations to find the value of the third variable. 6. Write your solution as an ordered triple. Observation: If a contradiction or an identity statement is found through the process of rewriting into triangular form, then the system has no solution or infinitely-many solutions, respectively.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 20 - of 45 Ex. 1 Solve the system of equations. x y 2z 2 3x y 5z 8 2x y 2z 7

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 21 - of 45 Ex. 2 Solve the system of equations. x y z0 3x y 6 x 2y 5z 3

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 22 - of 45 Ex. 3 Solve the system of equations. 2x y 3z 8 x y z 5 2x 4y 5z 1

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 23 - of 45 Topic 6: Intercepts; Symmetry of a Graph Ex. 1a Find the intercepts of 12x9y 36. 2 Definition: An intercept is a point of a graph on an axis. If you have x and y axes, the any points on these axes would be x-intercepts and y-intercepts, respectively. To find intercepts of an equation for one variable, set the other variable equal to zero and solve. Thus, for an equation involving x and y variables: To find x-intercepts, set y = 0 and solve for x To find y-intercepts, set x= 0 and solve for y x = h 1, x = h 2, x = h 3, y = k 1, y = k 2, y = k 3, (h 1, 0), (h 2, 0), (h 3, 0), (0, k 1 ), (0, k 2 ), (0, k 3 ),

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 24 - of 45 To determine if the graph of an equation displays symmetry, apply a test to see if the equation satisfies the definition of symmetry. To test for symmetry about the x-axis: Replace y y in the equation and simplify. If the original equation is returned, then (the graph of) the equation is symmetric about the x-axis. Ex. 1b Determine if the graph of 12x9y 36 displays any type of symmetry. 2 To test for symmetry about the y-axis: Replace x x in the equation and simplify. If the original equation is returned, then (the graph of) the equation is symmetric about the y-axis. To test for symmetry about the origin: Replace (x, y) with (, ) in the equation and simplify. If the original equation is returned, then the equation is symmetric about the origin.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 25 - of 45 Ex. 2 Find any intercepts for the equation 2 2x 4y 8and determine if its graph displays any type of symmetry.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 26 - of 45 Topic 7: An Introduction to Functions Visualizations of Functions Definition: A function is a relation defined by a rule that assigns each element in one set to exactly one element in a second set. The first set is called the domain. The domain of a function is the collection of values that can be acceptable inputs for a function. The variable of the domain is the independent variable of the function. The second set is called the range. The range of a function is the collection of values that can result from an input into the function. The variable of the range is the dependent variable of the function.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 27 - of 45 Functions can be expressed in many ways: Verbally As Data Visually As Equations A verbal rule that satisfies the definition A collection of points that satisfies the definition A graph representing a collection of points that satisfy the defintion A symbolic statement representing a rule satisfying the definition More often we run into relations expressed as equations or expressed visually and strategies are helpful to determine when a function exists. When a relation is expressed in graphical form, the following concept can be applied: Concept: The Vertical Line Test states that a graph represents a function if every possible vertical line intersects the graph at most once. Often it is obvious if a relation is a function when it is expressed verbally or conveyed as data.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 28 - of 45 Ex. 1 Determine if each graph of a relation below represents a function. Then determine the domain and range of the functions. Ex. 2 Determine if each graph of a relation below represents a function. Then determine the domain and range of the functions. a. a. b. b.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 29 - of 45 When a relation is expressed as an equation, the following concept can be applied: Ex. 4 Determine if each equation below represents a function. Concept: An equation represents a function if it can be uniquely solved for the variable of the range (often y). a. 2xy 8 Ex. 3 Determine if each equation below represents a function. 3 a. y x x 3 3 b. x y 1 2 b. x y 1

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 30 - of 45 Topic 8: Function Notation independent variable variable of the domain name of the function 2 f x x 3, x 2 rule of the function Ex. 2 Express the verbally defined function using function notation. a. Given x, multiply by 2, then add 4, and then take the square root. Ex. 1 dependent variable variable of the range Identify the parts of the functions given. a. g t 15 t, t 0 conditions of the function (may not always be present) b. Given x, where x is greater than or equal to 1, square and then subtract 2. b. Hx x c. Given x, take the square root, subtract 1, and then divide by 3.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 31 - of 45 Evaluating versus Solving Definition: Definition: Solve: 6x 7 10 To solve an equation, you want to find values of the variable(s) which make the equation true. To evaluate a function, you need a value for the independent variable and by applying the rule of the function, you find the value of the dependent variable associated with it. Functions can be evaluated for numerical values, constant values represented by letters, variables and variable expressions, or some other combination thereof. Ex. 3 Evaluate the function at the given values of x. f x 5x2 if x = x or if x = 2. a Evaluate: 2 f x x 3 for x = 1

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 32 - of 45 Ex. 4a Evaluate the function at the given values of x. Ex. 4b Evaluate the function at the given values of x. 2 f x x 3 if x = a 2. 2 f x x 3 if x = x + h.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 33 - of 45 A special kind of function which involves multiple rules and conditions for using each rule is called a piecewise-defined function. Example Ex. 5 4 x, x 2 gx ( ) 2x 3, x 2 Evaluate function g above if: a. x Ex. 6 Evaluate the piecewise-defined function given below at each value of x. 4 x, x 3 f( x) 5 x, x 3 a. x = 3 b. x = 4 b. x = 1

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 34 - of 45 Topic 9: Domains of Functions The domain of a function can either be explicitly or implicitly defined and it can either be limited or unlimited over the set of all real numbers. Most functions have an implicitly unlimited domain; that is, the domain of the function is the set of all real numbers. If a function has a condition attached to its rule, then its domain is explicitly defined and may be limited or unlimited. Some functions have implicitly limited domains. Despite lacking a condition stated with the rule, these functions are not defined for all real values of the independent variable. Two families of functions that are often implicitly limited: Radical Functions (with an even radical index) Example f x x x 2 3, 2 Rational Functions Functions which involve radicals and rational expressions can often be implicitly limited in multiple ways.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 35 - of 45 Ex. 1 Find the domain of f x 52x. Ex. 2 Find the domain of Fx x 6. 2 x 4

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 36 - of 45 Ex. 3 Find the domain of gx x 4 x 1. Ex. 4 Find the domain of Gx x 1. x 4

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 37 - of 45 Topic 10: Even and Odd Functions A function f is even if f x f x. A function f is odd if f x f x. Many functions are neither even nor odd. A basic example of an even function is f(x) = x² and a basic example of an odd function is g(x) = x³. Even functions display symmetry with respect to the y-axis (assuming f(x) = y) and odd functions display symmetry with respect to the origin. Because of the relationship between evenness/oddness and symmetry, the processes for determining each are somewhat similar. To determine if a function is either even or odd, evaluate the function at x and then simplify. 1. If it is possible to simplify back to the original function, then the function is even. 2. If it is possible to simplify to the opposite of the original function, then the function is odd. 3. If it not possible to return to either the original function or the opposite of the original, then the function is neither even nor odd.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 38 - of 45 Ex. 1 Determine if the function below is even, odd, or neither even nor odd. Ex. 2 Determine if the function below is even, odd, or neither even nor odd. 3 f x 4x x gx 4x 6x 3 x

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 39 - of 45 Ex. 3 Determine if the function below is even, odd, or neither even nor odd. Ex. 4 Determine if the function below is even, odd, or neither even nor odd. hx6x 5x 5 6 2 jx 4 x 5

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 40 - of 45 Topic 11: Graphs of Functions, a Summary Recall that the graph of a function is the collection of all the ordered pairs formed by evaluating the function. Recall that the y-intercept is the point of a graph of a function on the y-axis, if it exists. Recall that the x-intercept(s) are the point(s) of a graph of a function on the x-axis. For the purposes of the following definitions, assume that the first variable is x, the function is f, and the second variable f(x) = y: Recall that the domain is the collection of all possible values of x for the function. Recall that the range is the collection of all possible values of y for the function. The zeroes of the function are the values of x such that f(x) = 0. x-intercepts occur at zeroes. A function is said to be increasing over an interval if for all x 1 < x 2 in the interval, f(x 1 ) < f(x 2 ). A function is said to be decreasing over an interval if for all x 1 < x 2 in the interval, f(x 1 ) > f(x 2 ).

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 41 - of 45 A point where a function changes from increasing to decreasing, or vice versa, is called a turning point. Turning points are locations where extreme points occur. A point of a function (h, k) where k < f(x) for all x in an open interval containing h is called a relative minimum point of f. If k < f(x) for all x in the domain of f, then (h, k) is simply called the minimum point. A point of a function (h, k) where k > f(x) for all x in an open interval containing h is called a relative maximum point of f. If k > f(x) for all x in the domain of f, then (h, k) is simply called the maximum point. You may also see/hear relative extreme points referred to as local extreme points. A true extreme point is often referred to as an absolute extreme point or a global extreme point. Ex. Analyze the graph of f below.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 42 - of 45 Ex. 1 Identify and/or determine the properties of the function represented in the graph. Identify the intercepts of f. Evaluate f(2). Solve f(x) = 5. Identify the intervals over which f is increasing or decreasing. Identify the domain and range of f. Identify the extreme points of f.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 43 - of 45 Ex. 2 Identify and/or determine the properties of the function represented in the graph. Identify the intercepts of f. Evaluate f(2). Solve f(x) = 2. Identify the intervals over which f is increasing or decreasing. Identify the domain and range of f. Identify the extreme points of f.

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 44 - of 45 Topic 12: Graphing Piecewise-Defined Functions Here is a summary of our approach: 1. Set each rule equal to y. 2. Identify any boundary implied by the conditions and graph a dotted vertical line. 3. Evaluate each rule at any boundary that applies. Appropriately mark an open or closed dot with respect to whether the rule is used at the boundary. 4. Use appropriate techniques to graph each rule within its conditions. a. Rules that are constant will be horizontal segments or rays. b. Rules that are linear can be sketched using the slope starting at the point on the boundary c. Rules that are nonlinear can be sketched using a table of values or other appropriate methods. Evaluate the piecewise-defined function at x = 3 and then sketch a graph of the function. Ex. 1 4, x 1 f( x) x 2, x 1

Hartfield College Algebra (Version 2017a - Thomas Hartfield) Unit ONE Page - 45 - of 45 Evaluate the piecewise-defined function at x = 2 and then sketch a graph of the function. Evaluate the piecewise-defined function at x = 1 and then sketch a graph of the function. Ex. 2 1 x4, x 2 gx ( ) 2 3x2, x 2 Ex. 3 2x 3, x 2 hx ( ) 5 x, x 2