Basic Equations and Inequalities. An equation is a statement that the values of two expressions are equal.

Transcription

1 Hartfield College Algebra (Version Thomas Hartfield) Unit ZERO Page of 7 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 3x2 14x

2 Hartfield College Algebra (Version Thomas Hartfield) Unit ZERO Page of 7 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.

3 Hartfield College Algebra (Version Thomas Hartfield) Unit ZERO Page of 7 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

4 Hartfield College Algebra (Version Thomas Hartfield) Unit ZERO Page of 7 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 x 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 x 113 Ex. 5 x 4 2

5 Hartfield College Algebra (Version Thomas Hartfield) Unit ZERO Page of 7 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

6 Hartfield College Algebra (Version Thomas Hartfield) Unit ZERO Page of 7 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

7 Hartfield College Algebra (Version Thomas Hartfield) Unit ZERO Page of 7 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 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