Introduction to Inequalities - PDF Free Download

Introduction to Inequalities Statement of Prerequisite Skills Complete all previous TLM modules before beginning this module. Required Supporting Materials Access to the World Wide Web. Internet Eplorer 5.5 or greater. Macromedia Flash Plaer. Rationale Wh is it important for ou to learn this material? As ou ve seen, solving equations is a major part of mathematics. There are man situations, however, that require a solution to be greater than, less than, greater than or equal to, or less than or equal to a value rather than simpl being equal to a value. Learning Outcome When ou complete this module ou will be able to Solve and graph linear inequalities. Learning Objectives. Solve linear inequalities in one variable.. Graph linear inequalities in one variable.. Graph linear inequalities in two variables. Connection Activit Consider an electronic circuit that can sustain a maimum current of 00 milliamps. What set of potential currents could this circuit handle? Module A7 Introduction to Inequalities

OBJECTIVE ONE When ou complete this objective ou will be able to Solve linear inequalities in one variable. Eploration Activit Linear Inequalities An inequalit is a statement that one number is not equal to another number. The smbol > means greater than. The smbol < means less than. The smbol means greater than or equal to. The smbol means less than or equal EXAMPLE. > is read is greater than.. is read is less than or equal to.. a < < b is read is greater than a and less than b. If two inequalities have their inequalit smbols pointing in the same direction, then the two inequalities have the same sense. But, if the smbols point in opposite directions, the two inequalities are opposite in sense, or one is said to have the reverse sense of the other. For eample, the inequalities < and M < N have the same sense, but < and M > N have the opposite sense. An absolute inequalit is one that is true for all values of the variables. A conditional inequalit is one that is true for some of the values. EXAMPLE. 0 is an absolute inequalit because it is true for an real number.. > 0 is a conditional inequalit because it is true as long as 0. Module A7 Introduction to Inequalities

Eploration Activit Solving an Inequalit Solving an inequalit means finding all values of the variable which make the inequalit a true statement. The procedures are similar to solving linear equations. The following rules appl:. If the same number is added to or subtracted from both sides of the inequalit, then the new inequalit has the same sense as the original. For eample: If <, then + M < + M. If both sides of an inequalit are multiplied or divided b the same positive number, then the new inequalit has the same sense as the original. For eample: If >, then a > a. If both sides of an inequalit are multiplied or divided b the same negative number, then the resulting inequalit has the opposite sense of the original. For eample: If >, then ( a) < ( a) which becomes: a < a When these rules are applied we get an equivalent inequalit. Its solution is the same as the original inequalit. Definition: A linear inequalit in the variable is an inequalit that can be written: a + b < 0 (>,, could also be used) where a and b are constants. In the following eamples we will solve the given inequalit b appling these rules. Our purpose is to isolate the variable as we would do in a linear equation. The difference is that in a linear equation we have one solution and in a linear inequalit we ma have man solutions. Module A7 Introduction to Inequalities

EXAMPLE Solve the following inequalities:. > 9. 5. 9 SOLUTIONS:. for > 9, we divide b to obtain: > as our solution, i.e. all values greater than constitute our solution.. for 5, we divide b 5 to obtain: /5 as our solution, i.e. all values less than or equal to /5 constitute our solution.. for 9, we divide b to obtain: as our solution. NOTE: the change in sense. EXAMPLE Solve t 5 < t SOLUTION: We want to isolate the unknown t. Add 5 to both sides and subtract t from both sides to obtain: t t < + 5 which becomes t < t < is our solution Module A7 Introduction to Inequalities

EXAMPLE Solve ( + ) SOLUTION: Again we wish to isolate the unknown. Remove parentheses to obtain: + Now add to both sides and subtract from both sides, + (multipl through b ) 8 is our solution. Module A7 Introduction to Inequalities 5

Eperiential Activit One Solve the following inequalities:. > 8. 5 <. 0. 7 5. < R +. + 9 7. 7(5 ) 8. ( ) > ( 5) 9. < 0. + 7 8. + 7 5 7. > 8.. 5 M ( ) > M + > Show Me. 5. The amount of electrical current needed to operate a certain motor is more than 5 amperes but less than.8 amperes. If such motors are being used, and i represents the total number of amperes required, write an inequalit epression for this situation. Module A7 Introduction to Inequalities

COMMENT For questions ou receive from the computer please be aware of the following notations:. The smbol < = > is used to represent an one of the 5 possible choices.. Graphing questions refer to a list of graphs for the answer. Eperiential Activit One Answers. >. < 5. 0. 5. R >. 7 7. 7 8 8. > 9. < 8 0. 7.. < 0. M <. > 0 5. 55 < i < 7.8 Module A7 Introduction to Inequalities 7

OBJECTIVE TWO When ou complete this objective ou will be able to Graph linear inequalities in one variable. Eploration Activit The solutions of inequalities ma be plotted on a graph. For real number solutions we can plot them on a real line. The following notation is commonl used: a [ a, b] b a ( a, b) b For the top line a and b are contained in the solution. This is called a closed interval. It contains the endpoints of the interval. The square brackets mean the endpoints are included. For the bottom line a and b are not contained in the solution. This is called an open interval. The endpoints are not contained in the solution. The round brackets mean the endpoints are not included. Some additional eplanations follow: [ b) a, a < b a b (, ) a > a a (, a] a a ( b] a, a < b a b 8 Module A7 Introduction to Inequalities

Eperiential Activit Two Indicate the answers to odd numbered questions in Eperiential Activit on a real number line. Eperiential Activit Two Answers. 5 7 8 9 0 5 7 8. -7 - -5 - - - - 0 5 7 5. -7 - -5 - - - - 0 5 7 7. -7 - -5 - - - - 0 5 7 8 7 9. 8 9 0 5 7 8 9 0. -7 - -5 - - - - 0 5 7. -7 - -5 - - - - 0 5 7 Module A7 Introduction to Inequalities 9

OBJECTIVE THREE When ou complete this objective ou will be able to Graph linear inequalities in two variables. Eploration Activit A linear inequalit in the two variables M and N is an inequalit that can be written as am + bn + c < 0 ( or 0, 0, > 0) where a, b, and c are real constants with not both a and b equal to 0. Let us look at the solutions for an inequalit in two variables, i.e. > 5. We start with the graph of = 5 - - - ( 0, ) 5 - - 5 (,0) - This line = 5 separates the coordinate ais into sections:. The area above the line consisting of all (, ) points such < 5.. The area below the line consisting of all (, ) points such > 5.. The points (, ) on the line = 5. 0 Module A7 Introduction to Inequalities

EXAMPLE Solve ( ) ( ) + graphicall SOLUTION: Solving for we get 9 8 + 5 + 5 5 Now sketch the solution to 5 5 and we get the following graph: Note: When the graph includes the line then the line is solid. If the graph did not include the line it would be broken. - - - (,0) ( 0, 5) - ( ) ( ) + 5 5 - - The solution is all points in the area above the line and all points on the line. Module A7 Introduction to Inequalities

EXAMPLE Solve 5 + > 0( ) + graphicall. SOLUTION: Solve: 5 + > 0( ) + l for 5 + > 0 0+ 8 > 5 + 5 < 8 8 Now sketch the solution to < 5 8 8 0. 0. 0. Note: The line is broken because it is not included in the solution. -0. -0. -0. ) (0, 8 ( 0, ) 8-0. -0. 0. 0. 0. ( 5,0) 5 + > 0( ) + 5 < 8 8-0. 5 The solution consists of all points below the line = 8 8 Module A7 Introduction to Inequalities

EXAMPLE Solve < graphicall. SOLUTION: Because does not show up in the inequalit, the inequalit is assumed to be true for all values of. The solution consists of all points to the left of the line <. < (,0) Module A7 Introduction to Inequalities

Eperiential Activit Three Show the solutions for each of the following inequalities on a graph:. >. 5. <. > 5 5. ( + ) > ( ). > ( 5) 7. + 8. + ( ) > 7 9. + 7 < + ( + ) Show Me. 0. 5( + ) > ( ) Module A7 Introduction to Inequalities

Eperiential Activit Three Answers Please note that each graph has a comment above it that indicates which points are in the answer.. All points to the left of the line.. All points on and to the right of the line. (, ) ( 0,0) ( 5,0) ( 0, 5). All points above the line.. All points to the left of the line. ( 0, ) (,0) ( ) 5,0 Module A7 Introduction to Inequalities 5

5. All points above the line.. All points above the line. ( 0,0) (, ) ( 0,0) (,) 7. All points on and to the left of the line. 8. All points to the right of the line. ( 0,) (,0) ( 0, 7 ) ( 7,0 ) 5 9. All points to the right of the line. 0. All points to the right of the line. 8 ( 7 8,0) (,0) ( 0, 7) - 8 - Module A7 Introduction to Inequalities

Inequalities List.. 5 7 8 9 0 5 7 8-7 - -5 - - - - 0 5 7.. -7 - -5 - - - - 0 5 7-7 - -5 - - - - 0 5 7 5.. -7 - -5 - - - - 0 5 7-7 - -5 - - - - 0 5 7 7. 8. 0 5 7 8 9 0-7 - -5 - - - - 0 5 7 9. 0. -7 - -5 - - - - 0 5 7-7 - -5 - - - - 0 5 7.. -7 - -5 - - - - 0 5 7-7 - -5 - - - - 0 5 7.. -7 - -5 - - - - 0 5 7-7 - -5 - - - - 0 5 7 5.. -7 - -5 - - - - 0 5 7 0 5 7 8 9 0 7. 8. -7 - -5 - - - - 0 5 7-7 - -5 - - - - 0 5 7 9. 0. 5 7 8 9 0 5 7 8-7 - -5 - - - - 0 5 7.. -7 - -5 - - - - 0 5 7-7 - -5 - - - - 0 5 7.. 5. -7 - -5 - - - - 0 5 7-7 - -5 - - - - 0 5 7-7 - -5 - - - - 0 5 7

Inequalities List. All points on the line and below it.. All points on the line and below it. ( 0, ) (,0) - - ( 0, ) - (, 0 ) -. All points on the line and below it.. All points on the line and above it. - - (, 0 ) - ( 0, ) - - ( ) 0, - - - 5. All points on the line and below it.. All points on the line and above it. ( ) 0, (,0) (,0) ( 0, )

7. All points on the line and below it. 8. All points on the line and below it. - (,0) - ( ) 0, - - - ( 5,0) - ( 0, 5 ) - - - - 9. All points on the line and below it. 0. All points on the line and below it. ( 0, 5 ) ( 5,0) - - ( 0, ) - (, 0 ) -. All points on the line and below it.. All points on the line and below it. - ( ) 0, (,0) - ( 0, ) (,0 ) - -

. All points on the line and below it.. All points on the line and below it. ( 0, ) (,0) ( 0, ) (,0) 5. All points on the line and below it.. All points on the line and below it. ( 0, ) (,0) ( 5,0) ( 0, 5 ) 7. All points on the line and below it. 8. All points on the line and below it. ( 0, ) (,0) (,0) ( 0, )