Section 1.7 Inequalities

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Section 1.7 Inequalities"

Transcription

1 Section 1.7 Inequalities Linear Inequalities An inequality is linear if each term is constant or a multiple of the variable. EXAMPLE: Solve the inequality x < 9x+4 and sketch the solution set. x < 9x+4 x 9x < 9x+4 9x 6x < 4 6x 6 > 4 6 x > The solution set consists of all numbers that are greater than. In other words the solution of the inequality is the interval ( ),. EXAMPLE: Solve the inequality 5 6(x 4)+7 and sketch the solution set. 1

2 EXAMPLE: Solve the inequality 5 6(x 4)+7 and sketch the solution set. 5 6(x 4) (x 4) 1 6x x 1 6 6x 6 x The solution set consists of all numbers that are less than or equal to. In other words the solution of the inequality is the interval (,]. EXAMPLE: Solve the inequality 1 (x 6) 4 (x+) x and sketch the solution set. 4 1 (x 6) 4 (x+) 4 x ( 1 1 (x 6) 4 ) (x+) 1 ( 4 ) x 1 1 (x 6) 1 4 (x+) 1 ( 4 ) x 1 6(x 6) 16(x+) 9x 4 6x 6 16x 9x 4 6x 16x+9x 4+6+ x 44 ( 1)( x) ( 1)44 x 44 The solution set consists of all numbers that are less than or equal to 44. In other words the solution of the inequality is the interval (, 44]. -44

3 EXAMPLE: Solve the compound inequality 4 x < 1 and sketch the solution set. Therefore, the solution set is [, 5). 4 x < 1 4+ x + < 1+ 6 x < 15 6 x < 15 x < 5 EXAMPLE: Solve the compound inequality 1 < 5 x 4 1 < 5 x 4 4 ( 1) < 4 5 x < 5 x and sketch the solution set. 4 5 < 5 x < x 9 9 > x 9 > x 9 9 x < [ Therefore, the solution set is 9 ),. -9/

4 Nonlinear Inequalities EXAMPLE: Solve the inequality x 5x 6 and sketch the solution set. Solution: The corresponding equation x 5x+6 = (x )(x ) = 0 has the solutions and. As shown in the Figure below, the numbers and divide the real line into three intervals: (,), (,), and (, ). On each of these intervals we determine the signs of the factors using test values. We choose a number inside each interval and check the sign of the factors x and x at the value selected. For instance, if we use the test value x = 1 from the interval (,) shown in the Figure above, then substitution in the factors x and x gives x = 1 = 1, x = 1 = Both factors are negative on this interval, therefore x 5x+6 = (x )(x ) is positive on (,). Similarly, using the test values x = 1 and x = 4 from the intervals (,) and (, ), respectively, we get: + + Thus, the solution of the inequality x 5x 6 is {x x } = [,]. 4

5 EXAMPLE: Solve the inequality x(x 1)(x+) > 0 and sketch the solution set. Solution: The corresponding equation x(x 1)(x+) = 0 has the solutions 0,1, and. As shown in the Figure below, the numbers 0,1, and divide the real line into four intervals: (, ), (,0), (0,1), and (1, ) On each of these intervals we determine the signs of the factors using test values. We choose a number inside each interval and check the sign of the factors x, x 1, and x+ at the value selected. For instance, if we use the test value x = from the interval (, ) shown in the Figure above, then substitution in the factors x, x 1, and x+ gives x =, x 1 = 1 = 4, x+ = + = 1 All three factors are negative on this interval, therefore x(x 1)(x+) is negative on (, ). Similarly, using the test values x = 1, x = 1/ and x = from the intervals (,0), (0,1) and (1, ), respectively, we get: Thus, the solution of the inequality x(x 1)(x+) > 0 is (,0) (1, ). EXAMPLE: Solve the inequality x(x 1) (x ) < 0. Solution: The corresponding equation x(x 1) (x ) = 0 has the solutions 0,1, and. As shown in the Figure below, the numbers 0,1, and divide the real line into four intervals: (,0), (0,1), (1,), and (, ). 0 1 On each of these intervals we determine the signs of the factors using test values. We choose a number inside each interval and check the sign of the factors x and x at the value selected. For instance, if we use the test value x = 1 from the interval (,0) shown in the Figure above, then substitution in the factors x and x gives x = 1, x = 1 = 4 Bothfactorsarenegativeonthisinterval,(x 1) isalwaysnonnegative,thereforex(x 1) (x ) is positive on (,0). Similarly, using the test values x = 1/, x =, and x = 4 from the intervals (0, 1), (1, ), and (, ), respectively, we get: Thus, the solution of the inequality x(x 1) (x ) < 0 is (0,1) (1,). REMARK: The solution of the inequality x(x 1) (x ) 0 is [0,]. 5

6 EXAMPLE: Solve the following inequalities: (a) (x+)() 8 (b) (x+)(x+4) 1 (c) (x+)(x+4) 1 Solution: (a) We have (x+)() 8 x +4x+ 8 x +4x+ 8 0 x +4x 5 0 (x+5)(x 1) 0 The corresponding equation (x + 5)(x 1) = 0 has the solutions 5 and 1. As shown in the Figurebelow, thenumbers 5and1dividethereallineintothreeintervals: (, 5), ( 5,1), and (1, ) On each of these intervals we determine the signs of the factors using test values. We choose a number inside each interval and check the sign of the factors x + 5 and x 1 at the value selected. For instance, if we use the test value x = 6 from the interval (, 5) shown in the Figure above, then substitution in the factors x+5 and x 1 gives x+5 = 6+5 = 1, x 1 = 6 1 = 7 Both factors are negative on this interval, therefore (x+5)(x 1) is positive on (,5). Similarly, using the test values x = 0 and x = from the intervals ( 5,1) and (1, ), respectively, we get: Thus, the solution of the inequality (x+)() 8 is {x x 5 or x 1} = (, 5] [1, ). (b, c) We have (x+)(x+4) 1 x +6x+8 1 x +6x x } +6x+9 {{} 0 x + x + (x+) 0 Note that (x + ) is either positive or zero. Therefore, the only solution of the inequality (x + )(x + 4) 1 is x =. In a similar way one can show that the solution of the inequality (x+)(x+4) 1 is all real numbers. 6

7 EXAMPLE: Solve the inequality 1+x 1 and sketch the solution set. 1+x 1 1+x x 0 1+x () 1+x 1+x 0 0 x 0 As shown in the Figure below, the numbers 0 (at which the numerator of which the denominator of and (1, ). x x is 0) and 1 (at is 0) divide the real line into three intervals: (,0), (0,1), 0 1 x On each of these intervals we determine the sign of using test values. We choose a x number inside each interval and check the sign of at the value selected. For instance, if we use the test value x = 1 from the interval (,0) shown in the Figure above, then substitution in x gives ( 1) 1 ( 1) = = 1 < 0 and x = from the intervals (0,1), and (1, ), respec- Similarly, using the test values x = 1 tively, we get: Thus, the solution of the inequality 1+x 1 is [0,1). EXAMPLE: Solve the inequality x 1 and sketch the solution set. 7

8 EXAMPLE: Solve the inequality x 1 and sketch the solution set. x 1 x 1 0 x 0 x () x x x 4 0 As shown in the Figure below, the numbers 4 (at which the numerator of x 4 is 0) and 1 (at whichthedenominatorof x 4 is0)dividethereallineintothreeintervals: (, 1), ( 1,4), and (4, ) On each of these intervals we determine the sign of x 4 using test values. We choose a number inside each interval and check the sign of x 4 at the value selected. For instance, if we use the test value x = from the interval (, 1) shown in the Figure above, then substitution in x 4 gives 4 +1 = 6 1 = 6 > 0 Similarly, using the test values x = 0 and x = 5 from the intervals ( 1,4), and (4, ), respectively, we get: Thus, the solution of the inequality x 1 is ( 1,4]. EXAMPLE: Solve the inequality 7 5x 8x+9 < and sketch the solution set. 8

9 EXAMPLE: Solve the inequality 7 5x 8x+9 < and sketch the solution set. 7 5x 8x+9 < 7 5x 8x+9 < 0 (7 5x) (8x+9) < 0 (7 5x) (8x+9) 1 15x 16x 18 < 0 < 0 < 0 As shown in the Figure below, the numbers /1 (at which the numerator of is 0) and 9/8 (at which the denominator of is 0) divide the real line into three intervals: (, 9/8), ( 9/8, /1), and (/1, ). -9/8 /1 On each of these intervals we determine the sign of using test values. We choose a number inside each interval and check the sign of at the value selected. For instance, if we use the test value x = from the interval (, 9/8) shown in the Figure above, then substitution in gives = 1( ) (8( )+9) = +6 ( 16+9) = 65 ( 7) < 0 Similarly, using the test values x = 0 and x = 1 from the intervals ( 9/8,/1), and (/1, ), respectively, we get: -9/8 + /1 Thus, the solution of the inequality 7 5x 8x+9 < ( is, 9 ) ( ) 8 1,. -9/8 /1 9

10 Absolute Value Inequalities EXAMPLE: Solve the inequality x 5 < and sketch the solution set. Solution: The inequality x 5 < is equivalent to < x 5 < +5 < x 5+5 < +5 < x < 7 The solution set is the open interval (,7). EXAMPLE: Solve the inequality x 5 <. Solution: The inequality x 5 < has no solutions, since is always nonnegative. So, the solution set is the empty set. EXAMPLE: Solve the inequality 7 x < 1. 10

11 EXAMPLE: Solve the inequality 7 x < 1. Solution: The inequality 7 x < 1 is equivalent to The solution set is the open interval 1 < 7 x < < 7 x 7 < < x < 6 8 > x > 6 < x < 8 (, 8 ). EXAMPLE: Solve the inequality x+ 4 and sketch the solution set. Solution: The inequality x+ 4 is equivalent to x+ 4 or x+ 4 x x 6 x x The solution set is { x x or x } [ ) = (, ], EXAMPLE: Solve the inequality 4 5x x x+9 +7 }{{} 9 The inequality 5x is equivalent to 5x The solution set is { x 9 4 x x ( 4 9 ) 4 (5x+9) x x x = 45 0 x 7 0 } [ = 9 4, ].

If a product or quotient has an even number of negative factors, then its value is positive.

If a product or quotient has an even number of negative factors, then its value is positive. MATH 0 SOLVING NONLINEAR INEQUALITIES KSU Important Properties: If a product or quotient has an even number of negative factors, then its value is positive. If a product or quotient has an odd number of

More information

Absolute Value Equations and Inequalities

Absolute Value Equations and Inequalities . Absolute Value Equations and Inequalities. OBJECTIVES 1. Solve an absolute value equation in one variable. Solve an absolute value inequality in one variable NOTE Technically we mean the distance between

More information

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality.

Definition 8.1 Two inequalities are equivalent if they have the same solution set. Add or Subtract the same value on both sides of the inequality. 8 Inequalities Concepts: Equivalent Inequalities Linear and Nonlinear Inequalities Absolute Value Inequalities (Sections 4.6 and 1.1) 8.1 Equivalent Inequalities Definition 8.1 Two inequalities are equivalent

More information

Math 407A: Linear Optimization

Math 407A: Linear Optimization Math 407A: Linear Optimization Lecture 4: LP Standard Form 1 1 Author: James Burke, University of Washington LPs in Standard Form Minimization maximization Linear equations to linear inequalities Lower

More information

Chapter 1 Section 5: Equations and Inequalities involving Absolute Value

Chapter 1 Section 5: Equations and Inequalities involving Absolute Value Introduction The concept of absolute value is very strongly connected to the concept of distance. The absolute value of a number is that number s distance from 0 on the number line. Since distance is always

More information

1.1 Solving a Linear Equation ax + b = 0

1.1 Solving a Linear Equation ax + b = 0 1.1 Solving a Linear Equation ax + b = 0 To solve an equation ax + b = 0 : (i) move b to the other side (subtract b from both sides) (ii) divide both sides by a Example: Solve x = 0 (i) x- = 0 x = (ii)

More information

Inequalities - Absolute Value Inequalities

Inequalities - Absolute Value Inequalities 3.3 Inequalities - Absolute Value Inequalities Objective: Solve, graph and give interval notation for the solution to inequalities with absolute values. When an inequality has an absolute value we will

More information

Definition: Absolute Value The absolute value of a number is the distance that the number is from zero. The absolute value of x is written x.

Definition: Absolute Value The absolute value of a number is the distance that the number is from zero. The absolute value of x is written x. R Absolute Values We begin this section by recalling the following definition Definition: Absolute Value The absolute value of a number is the distance that the number is from zero The absolute value of

More information

Guide to SRW Section 1.7: Solving inequalities

Guide to SRW Section 1.7: Solving inequalities Guide to SRW Section 1.7: Solving inequalities When you solve the equation x 2 = 9, the answer is written as two very simple equations: x = 3 (or) x = 3 The diagram of the solution is -6-5 -4-3 -2-1 0

More information

LINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to,

LINEAR INEQUALITIES. less than, < 2x + 5 x 3 less than or equal to, greater than, > 3x 2 x 6 greater than or equal to, LINEAR INEQUALITIES When we use the equal sign in an equation we are stating that both sides of the equation are equal to each other. In an inequality, we are stating that both sides of the equation are

More information

Absolute Value Equations and Inequalities

Absolute Value Equations and Inequalities Key Concepts: Compound Inequalities Absolute Value Equations and Inequalities Intersections and unions Suppose that A and B are two sets of numbers. The intersection of A and B is the set of all numbers

More information

7.5 SYSTEMS OF INEQUALITIES. Copyright Cengage Learning. All rights reserved.

7.5 SYSTEMS OF INEQUALITIES. Copyright Cengage Learning. All rights reserved. 7.5 SYSTEMS OF INEQUALITIES Copyright Cengage Learning. All rights reserved. What You Should Learn Sketch the graphs of inequalities in two variables. Solve systems of inequalities. Use systems of inequalities

More information

Chapter 2: Linear Equations and Inequalities Lecture notes Math 1010

Chapter 2: Linear Equations and Inequalities Lecture notes Math 1010 Section 2.1: Linear Equations Definition of equation An equation is a statement that equates two algebraic expressions. Solving an equation involving a variable means finding all values of the variable

More information

STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS

STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS STUDY GUIDE FOR SOME BASIC INTERMEDIATE ALGEBRA SKILLS The intermediate algebra skills illustrated here will be used extensively and regularly throughout the semester Thus, mastering these skills is an

More information

Determining When an Expression Is Undefined

Determining When an Expression Is Undefined Determining When an Expression Is Undefined Connections Have you ever... Tried to use a calculator to divide by zero and gotten an error Tried to figure out the square root of a negative number Expressions

More information

Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2

Warm Up Lesson Presentation Lesson Quiz. Holt Algebra 2 2 2-8 Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Solve. 1. y + 7 < 11 2. 4m 12 3. 5 2x 17 y < 18 m 3 x 6 Use interval notation to indicate the graphed numbers. 4. (-2, 3] 5. (-, 1] Objectives Solve

More information

Math 1111 Journal Entries Unit I (Sections , )

Math 1111 Journal Entries Unit I (Sections , ) Math 1111 Journal Entries Unit I (Sections 1.1-1.2, 1.4-1.6) Name Respond to each item, giving sufficient detail. You may handwrite your responses with neat penmanship. Your portfolio should be a collection

More information

Quadratic Equations and Inequalities

Quadratic Equations and Inequalities MA 134 Lecture Notes August 20, 2012 Introduction The purpose of this lecture is to... Introduction The purpose of this lecture is to... Learn about different types of equations Introduction The purpose

More information

Solving and Graphing Compound Inequalities of a Single Variable

Solving and Graphing Compound Inequalities of a Single Variable Section 3.2 Chapter 3 Graphing Fundamentals Solving and Graphing Compound Inequalities of a Single Variable TERMINOLOGY 3.2 Prerequisite Terms: Absolute Value Inequality Your definition New Terms to Learn:

More information

Section 3.7 Rational Functions

Section 3.7 Rational Functions Section 3.7 Rational Functions A rational function is a function of the form where P and Q are polynomials. r(x) = P(x) Q(x) Rational Functions and Asymptotes The domain of a rational function consists

More information

Algebra. Indiana Standards 1 ST 6 WEEKS

Algebra. Indiana Standards 1 ST 6 WEEKS Chapter 1 Lessons Indiana Standards - 1-1 Variables and Expressions - 1-2 Order of Operations and Evaluating Expressions - 1-3 Real Numbers and the Number Line - 1-4 Properties of Real Numbers - 1-5 Adding

More information

ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section

ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 5-3.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 6-4.2 Solving Equations by

More information

Review of Key Concepts: 1.2 Characteristics of Polynomial Functions

Review of Key Concepts: 1.2 Characteristics of Polynomial Functions Review of Key Concepts: 1.2 Characteristics of Polynomial Functions Polynomial functions of the same degree have similar characteristics The degree and leading coefficient of the equation of the polynomial

More information

Solving Quadratic & Higher Degree Inequalities

Solving Quadratic & Higher Degree Inequalities Ch. 8 Solving Quadratic & Higher Degree Inequalities We solve quadratic and higher degree inequalities very much like we solve quadratic and higher degree equations. One method we often use to solve quadratic

More information

5 =5. Since 5 > 0 Since 4 7 < 0 Since 0 0

5 =5. Since 5 > 0 Since 4 7 < 0 Since 0 0 a p p e n d i x e ABSOLUTE VALUE ABSOLUTE VALUE E.1 definition. The absolute value or magnitude of a real number a is denoted by a and is defined by { a if a 0 a = a if a

More information

CHAPTER 3: Quadratic Functions and Equations; Inequalities

CHAPTER 3: Quadratic Functions and Equations; Inequalities MAT 171 Precalculus Algebra Dr. Claude Moore Cape Fear Community College CHAPTER 3: Quadratic Functions and Equations; Inequalities 3.1 The Complex Numbers 3.2 Quadratic Equations, Functions, Zeros, and

More information

Practice Math Placement Exam

Practice Math Placement Exam Practice Math Placement Exam The following are problems like those on the Mansfield University Math Placement Exam. You must pass this test or take MA 0090 before taking any mathematics courses. 1. What

More information

Isolate the absolute value expression. Add 5 to both sides and then divide by 2.

Isolate the absolute value expression. Add 5 to both sides and then divide by 2. 11 of 21 8/14/2014 2:35 PM of a variable expression. You can use the definition of absolute value to solve absolute value equations algebraically. Since then the equation ax + b = c is equivalent to (ax

More information

Chapter 2 Test Review

Chapter 2 Test Review Name Chapter 2 Test Review MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. If the following is a polynomial function, then state its degree and leading

More information

7. Solving Linear Inequalities and Compound Inequalities

7. Solving Linear Inequalities and Compound Inequalities 7. Solving Linear Inequalities and Compound Inequalities Steps for solving linear inequalities are very similar to the steps for solving linear equations. The big differences are multiplying and dividing

More information

Ordered Pairs. Graphing Lines and Linear Inequalities, Solving System of Linear Equations. Cartesian Coordinates System.

Ordered Pairs. Graphing Lines and Linear Inequalities, Solving System of Linear Equations. Cartesian Coordinates System. Ordered Pairs Graphing Lines and Linear Inequalities, Solving System of Linear Equations Peter Lo All equations in two variables, such as y = mx + c, is satisfied only if we find a value of x and a value

More information

This is a square root. The number under the radical is 9. (An asterisk * means multiply.)

This is a square root. The number under the radical is 9. (An asterisk * means multiply.) Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize

More information

Algebraic Simplex Method - Introduction Previous

Algebraic Simplex Method - Introduction Previous Algebraic Simplex Method - Introduction To demonstrate the simplex method, consider the following linear programming model: This is the model for Leo Coco's problem presented in the demo, Graphical Method.

More information

Chapter R - Basic Algebra Operations (69 topics, due on 05/01/12)

Chapter R - Basic Algebra Operations (69 topics, due on 05/01/12) Course Name: College Algebra 001 Course Code: R3RK6-CTKHJ ALEKS Course: College Algebra with Trigonometry Instructor: Prof. Bozyk Course Dates: Begin: 01/17/2012 End: 05/04/2012 Course Content: 288 topics

More information

Lesson 7: Solving Absolute Value Inequalities. Absolute Value Inequalities with Less Than (<)

Lesson 7: Solving Absolute Value Inequalities. Absolute Value Inequalities with Less Than (<) Lesson 7: Solving Absolute Value Absolute Value with Less Than () x > 5 For all real numbers, a and b, if b>0, then the following statements are true. 1. If

More information

Directions Please read carefully!

Directions Please read carefully! Math Xa Algebra Practice Problems (Solutions) Fall 2008 Directions Please read carefully! You will not be allowed to use a calculator or any other aids on the Algebra Pre-Test or Post-Test. Be sure to

More information

Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities

Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities Solving Systems of Equations with Absolute Value, Polynomials, and Inequalities Solving systems of equations with inequalities When solving systems of linear equations, we are looking for the ordered pair

More information

Absolute Value Equations

Absolute Value Equations Absolute Value Equations Discussion: Absolute value refers to the measure of distance from zero f any value on the number line. F example, the absolute value of 3 is 3 (written as ) because there are three

More information

Sect Solving Absolute Values Inequalities. Solving Absolute Value Inequalities using the Definition

Sect Solving Absolute Values Inequalities. Solving Absolute Value Inequalities using the Definition 17 Sect 10.4 - Solving Absolute Values Inequalities Concept #1 Solving Absolute Value Inequalities using the Definition In the last section, we saw that the solution to x = 4 was x = 4 and x = 4. If we

More information

Solving Equations with One Variable Type - The Algebraic Approach. Solving Equations with a Variable in the Denominator - The Algebraic Approach

Solving Equations with One Variable Type - The Algebraic Approach. Solving Equations with a Variable in the Denominator - The Algebraic Approach 3 Solving Equations Concepts: Number Lines The Definitions of Absolute Value Equivalent Equations Solving Equations with One Variable Type - The Algebraic Approach Solving Equations with a Variable in

More information

Review for Calculus Rational Functions, Logarithms & Exponentials

Review for Calculus Rational Functions, Logarithms & Exponentials Definition and Domain of Rational Functions A rational function is defined as the quotient of two polynomial functions. F(x) = P(x) / Q(x) The domain of F is the set of all real numbers except those for

More information

7.4 Linear Programming: The Simplex Method

7.4 Linear Programming: The Simplex Method 7.4 Linear Programming: The Simplex Method For linear programming problems with more than two variables, the graphical method is usually impossible, so the simplex method is used. Because the simplex method

More information

1.1. Basic Concepts. Write sets using set notation. Write sets using set notation. Write sets using set notation. Write sets using set notation.

1.1. Basic Concepts. Write sets using set notation. Write sets using set notation. Write sets using set notation. Write sets using set notation. 1.1 Basic Concepts Write sets using set notation. Objectives A set is a collection of objects called the elements or members of the set. 1 2 3 4 5 6 7 Write sets using set notation. Use number lines. Know

More information

Algebra II Pacing Guide First Nine Weeks

Algebra II Pacing Guide First Nine Weeks First Nine Weeks SOL Topic Blocks.4 Place the following sets of numbers in a hierarchy of subsets: complex, pure imaginary, real, rational, irrational, integers, whole and natural. 7. Recognize that the

More information

Letter to the Student... 5 Letter to the Family... 6 Correlation of Mississippi Competencies and Objectives to Coach Lessons... 7 Pretest...

Letter to the Student... 5 Letter to the Family... 6 Correlation of Mississippi Competencies and Objectives to Coach Lessons... 7 Pretest... Table of Contents Letter to the Student........................................... 5 Letter to the Family............................................. 6 Correlation of Mississippi Competencies and Objectives

More information

2.3. Finding polynomial functions. An Introduction:

2.3. Finding polynomial functions. An Introduction: 2.3. Finding polynomial functions. An Introduction: As is usually the case when learning a new concept in mathematics, the new concept is the reverse of the previous one. Remember how you first learned

More information

10.9 Systems Of Inequalities. Copyright Cengage Learning. All rights reserved.

10.9 Systems Of Inequalities. Copyright Cengage Learning. All rights reserved. 10.9 Systems Of Inequalities Copyright Cengage Learning. All rights reserved. Objectives Graphing an Inequality Systems of Inequalities Systems of Linear Inequalities Application: Feasible Regions 2 Graphing

More information

Lecture 7 : Inequalities 2.5

Lecture 7 : Inequalities 2.5 3 Lecture 7 : Inequalities.5 Sometimes a problem may require us to find all numbers which satisfy an inequality. An inequality is written like an equation, except the equals sign is replaced by one of

More information

3y 1 + 5y 2. y 1 + y 2 20 y 1 0, y 2 0.

3y 1 + 5y 2. y 1 + y 2 20 y 1 0, y 2 0. 1 Linear Programming A linear programming problem is the problem of maximizing (or minimizing) a linear function subject to linear constraints. The constraints may be equalities or inequalities. 1.1 Example

More information

2. INEQUALITIES AND ABSOLUTE VALUES

2. INEQUALITIES AND ABSOLUTE VALUES 2. INEQUALITIES AND ABSOLUTE VALUES 2.1. The Ordering of the Real Numbers In addition to the arithmetic structure of the real numbers there is the order structure. The real numbers can be represented by

More information

Algebra Revision Sheet Questions 2 and 3 of Paper 1

Algebra Revision Sheet Questions 2 and 3 of Paper 1 Algebra Revision Sheet Questions and of Paper Simple Equations Step Get rid of brackets or fractions Step Take the x s to one side of the equals sign and the numbers to the other (remember to change the

More information

ALGEBRA I / ALGEBRA I SUPPORT

ALGEBRA I / ALGEBRA I SUPPORT Suggested Sequence: CONCEPT MAP ALGEBRA I / ALGEBRA I SUPPORT August 2011 1. Foundations for Algebra 2. Solving Equations 3. Solving Inequalities 4. An Introduction to Functions 5. Linear Functions 6.

More information

Main page. Given f ( x, y) = c we differentiate with respect to x so that

Main page. Given f ( x, y) = c we differentiate with respect to x so that Further Calculus Implicit differentiation Parametric differentiation Related rates of change Small variations and linear approximations Stationary points Curve sketching - asymptotes Curve sketching the

More information

MAT 1033 Intermediate Algebra FINAL EXAM REVIEW PACKET

MAT 1033 Intermediate Algebra FINAL EXAM REVIEW PACKET MAT 1033 Intermediate Algebra FINAL EXAM REVIEW PACKET Solve the inequality. Write the solution using interval notation and graph on a number line. 1) 21x + 12 > 3(6x - 2) A) (-6, ~) -11-10 -9-8 -7 -~

More information

x 2 k S. S. k, k x 2 bx b 2 x b b2 4ac 2a b 2 4ac

x 2 k S. S. k, k x 2 bx b 2 x b b2 4ac 2a b 2 4ac Solving Quadratic Equations a b c 0, a 0 Methods for solving: 1. B factoring. A. First, put the equation in standard form. B. Then factor the left side C. Set each factor 0 D. Solve each equation. B square

More information

2.2. Using Second Derivatives to Find Maximum and Minimum Values and Sketch Graphs

2.2. Using Second Derivatives to Find Maximum and Minimum Values and Sketch Graphs Slide. - 1. Using Second Derivatives to Find Maximum and Minimum Values and Sketch Graphs OBJECTIVES Find the relative extrema of a function using the Second-Derivative Test. Sketch the graph of a continuous

More information

Solving inequalities. Jackie Nicholas Jacquie Hargreaves Janet Hunter

Solving inequalities. Jackie Nicholas Jacquie Hargreaves Janet Hunter Mathematics Learning Centre Solving inequalities Jackie Nicholas Jacquie Hargreaves Janet Hunter c 6 Universit of Sdne Mathematics Learning Centre, Universit of Sdne Solving inequalities In these nots

More information

Linear Programming Notes V Problem Transformations

Linear Programming Notes V Problem Transformations Linear Programming Notes V Problem Transformations 1 Introduction Any linear programming problem can be rewritten in either of two standard forms. In the first form, the objective is to maximize, the material

More information

Section 4.1 Inequalities & Applications. Inequalities. Equations. 3x + 7 = 13 y = 7 3x + 2y = 6. 3x + 7 < 13 y > 7 3x + 2y 6. Symbols: < > 4.

Section 4.1 Inequalities & Applications. Inequalities. Equations. 3x + 7 = 13 y = 7 3x + 2y = 6. 3x + 7 < 13 y > 7 3x + 2y 6. Symbols: < > 4. Section 4.1 Inequalities & Applications Equations 3x + 7 = 13 y = 7 3x + 2y = 6 Inequalities 3x + 7 < 13 y > 7 3x + 2y 6 Symbols: < > 4.1 1 Overview of Linear Inequalities 4.1 Study Inequalities with One

More information

6 Rational Inequalities, (In)equalities with Absolute value; Exponents and Logarithms

6 Rational Inequalities, (In)equalities with Absolute value; Exponents and Logarithms AAU - Business Mathematics I Lecture #6, March 16, 2009 6 Rational Inequalities, (In)equalities with Absolute value; Exponents and Logarithms 6.1 Rational Inequalities: x + 1 x 3 > 1, x + 1 x 2 3x + 5

More information

First Degree Equations First degree equations contain variable terms to the first power and constants.

First Degree Equations First degree equations contain variable terms to the first power and constants. Section 4 7: Solving 2nd Degree Equations First Degree Equations First degree equations contain variable terms to the first power and constants. 2x 6 = 14 2x + 3 = 4x 15 First Degree Equations are solved

More information

Course Name: Course Code: ALEKS Course: Instructor: Course Dates: Course Content: Textbook: Dates Objective Prerequisite Topics

Course Name: Course Code: ALEKS Course: Instructor: Course Dates: Course Content: Textbook: Dates Objective Prerequisite Topics Course Name: MATH 1204 Fall 2015 Course Code: N/A ALEKS Course: College Algebra Instructor: Master Templates Course Dates: Begin: 08/22/2015 End: 12/19/2015 Course Content: 271 Topics (261 goal + 10 prerequisite)

More information

Math 113 HW #10 Solutions

Math 113 HW #10 Solutions Math HW #0 Solutions. Exercise 4.5.4. Use the guidelines of this section to sketch the curve Answer: Using the quotient rule, y = x x + 9. y = (x + 9)(x) x (x) (x + 9) = 8x (x + 9). Since the denominator

More information

Math 113 HW #9 Solutions

Math 113 HW #9 Solutions Math 3 HW #9 Solutions 4. 50. Find the absolute maximum and absolute minimum values of on the interval [, 4]. f(x) = x 3 6x 2 + 9x + 2 Answer: First, we find the critical points of f. To do so, take the

More information

MA107 Precalculus Algebra Exam 2 Review Solutions

MA107 Precalculus Algebra Exam 2 Review Solutions MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write

More information

Maximum and Minimum Values

Maximum and Minimum Values Jim Lambers MAT 280 Spring Semester 2009-10 Lecture 8 Notes These notes correspond to Section 11.7 in Stewart and Section 3.3 in Marsden and Tromba. Maximum and Minimum Values In single-variable calculus,

More information

MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions

MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions MSLC Workshop Series Math 1148 1150 Workshop: Polynomial & Rational Functions The goal of this workshop is to familiarize you with similarities and differences in both the graphing and expression of polynomial

More information

Solving Quadratic Equations by Completing the Square

Solving Quadratic Equations by Completing the Square 9. Solving Quadratic Equations by Completing the Square 9. OBJECTIVES 1. Solve a quadratic equation by the square root method. Solve a quadratic equation by completing the square. Solve a geometric application

More information

Rational inequality. Sunil Kumar Singh. 1 Sign scheme or diagram for rational function

Rational inequality. Sunil Kumar Singh. 1 Sign scheme or diagram for rational function OpenStax-CNX module: m15464 1 Rational inequality Sunil Kumar Singh This work is produced by OpenStax-CNX and licensed under the Creative Commons Attribution License 2.0 Rational inequality is an inequality

More information

Contents. The Real Numbers. Linear Equations and Inequalities in One Variable

Contents. The Real Numbers. Linear Equations and Inequalities in One Variable dug33513_fm.qxd 11/20/07 3:21 PM Page vii Preface Guided Tour: Features and Supplements Applications Index 1 2 The Real Numbers 1.1 1.2 1.3 1.4 1.5 1.6 1 Sets 2 The Real Numbers 9 Operations on the Set

More information

is the degree of the polynomial and is the leading coefficient.

is the degree of the polynomial and is the leading coefficient. Property: T. Hrubik-Vulanovic e-mail: thrubik@kent.edu Content (in order sections were covered from the book): Chapter 6 Higher-Degree Polynomial Functions... 1 Section 6.1 Higher-Degree Polynomial Functions...

More information

Calculus Card Matching

Calculus Card Matching Card Matching Card Matching A Game of Matching Functions Description Give each group of students a packet of cards. Students work as a group to match the cards, by thinking about their card and what information

More information

McMurry University Pre-test Practice Exam. 1. Simplify each expression, and eliminate any negative exponent(s).

McMurry University Pre-test Practice Exam. 1. Simplify each expression, and eliminate any negative exponent(s). 1. Simplify each expression, and eliminate any negative exponent(s). a. b. c. 2. Simplify the expression. Assume that a and b denote any real numbers. (Assume that a denotes a positive number.) 3. Find

More information

Inequalities and Absolute Value Equations and Inequations

Inequalities and Absolute Value Equations and Inequations Inequalities and Absolute Value Equations and Inequations - 2.4-2.5 Fall 2013 - Math 1010 A no TIE fighter and squinting cat zone. (Math 1010) M 1010 2.4-2.4 1 / 12 Roadmap Notes for solving inequalities.

More information

Section 1.1 Real Numbers

Section 1.1 Real Numbers . Natural numbers (N):. Integer numbers (Z): Section. Real Numbers Types of Real Numbers,, 3, 4,,... 0, ±, ±, ±3, ±4, ±,... REMARK: Any natural number is an integer number, but not any integer number is

More information

Chapter 6 Notes. Section 6.1 Solving One-Step Linear Inequalities

Chapter 6 Notes. Section 6.1 Solving One-Step Linear Inequalities Chapter 6 Notes Name Section 6.1 Solving One-Step Linear Inequalities Graph of a linear Inequality- the set of all points on a number line that represent all solutions of the inequality > or < or circle

More information

High School Mathematics Algebra

High School Mathematics Algebra High School Mathematics Algebra This course is designed to give students the foundation of understanding algebra at a moderate pace. Essential material will be covered to prepare the students for Geometry.

More information

Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4)

Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4) Chapter 2: Functions and Linear Functions 1. Know the definition of a relation. Math 155 (DoVan) Exam 1 Review (Sections 3.1, 3.2, 5.1, 5.2, Chapters 2 & 4) 2. Know the definition of a function. 3. What

More information

Identify examples of field properties: commutative, associative, identity, inverse, and distributive.

Identify examples of field properties: commutative, associative, identity, inverse, and distributive. Topic: Expressions and Operations ALGEBRA II - STANDARD AII.1 The student will identify field properties, axioms of equality and inequality, and properties of order that are valid for the set of real numbers

More information

Algebra Course KUD. Green Highlight - Incorporate notation in class, with understanding that not tested on

Algebra Course KUD. Green Highlight - Incorporate notation in class, with understanding that not tested on Algebra Course KUD Yellow Highlight Need to address in Seminar Green Highlight - Incorporate notation in class, with understanding that not tested on Blue Highlight Be sure to teach in class Postive and

More information

COGNITIVE TUTOR ALGEBRA

COGNITIVE TUTOR ALGEBRA COGNITIVE TUTOR ALGEBRA Numbers and Operations Standard: Understands and applies concepts of numbers and operations Power 1: Understands numbers, ways of representing numbers, relationships among numbers,

More information

Partial Fractions. Combining fractions over a common denominator is a familiar operation from algebra:

Partial Fractions. Combining fractions over a common denominator is a familiar operation from algebra: Partial Fractions Combining fractions over a common denominator is a familiar operation from algebra: From the standpoint of integration, the left side of Equation 1 would be much easier to work with than

More information

6.4 Logarithmic Equations and Inequalities

6.4 Logarithmic Equations and Inequalities 6.4 Logarithmic Equations and Inequalities 459 6.4 Logarithmic Equations and Inequalities In Section 6.3 we solved equations and inequalities involving exponential functions using one of two basic strategies.

More information

2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses

2. Simplify. College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key. Use the distributive property to remove the parentheses College Algebra Student Self-Assessment of Mathematics (SSAM) Answer Key 1. Multiply 2 3 5 1 Use the distributive property to remove the parentheses 2 3 5 1 2 25 21 3 35 31 2 10 2 3 15 3 2 13 2 15 3 2

More information

Appendix A: Numbers, Inequalities, and Absolute Values. Outline

Appendix A: Numbers, Inequalities, and Absolute Values. Outline Appendix A: Numbers, Inequalities, and Absolute Values Tom Lewis Fall Semester 2015 Outline Types of numbers Notation for intervals Inequalities Absolute value A hierarchy of numbers Whole numbers 1, 2,

More information

Finding Inverse Functions Guided Notes. Example 1: Find the inverse of a(b) = 3b 7

Finding Inverse Functions Guided Notes. Example 1: Find the inverse of a(b) = 3b 7 Inverse Functions Finding Inverse Functions Guided Notes We ve explored the idea that the inverse of a function undoes the function. One way of finding the inverse of a function is to. You can think of

More information

Lecture 4 Linear Programming Models: Standard Form. August 31, 2009

Lecture 4 Linear Programming Models: Standard Form. August 31, 2009 Linear Programming Models: Standard Form August 31, 2009 Outline: Lecture 4 Standard form LP Transforming the LP problem to standard form Basic solutions of standard LP problem Operations Research Methods

More information

Method To Solve Linear, Polynomial, or Absolute Value Inequalities:

Method To Solve Linear, Polynomial, or Absolute Value Inequalities: Solving Inequalities An inequality is the result of replacing the = sign in an equation with ,, or. For example, 3x 2 < 7 is a linear inequality. We call it linear because if the < were replaced with

More information

Chapter 2 Analysis of Graphs of Functions

Chapter 2 Analysis of Graphs of Functions Chapter Analysis o Graphs o Functions Chapter Analysis o Graphs o Functions Covered in this Chapter:.1 Graphs o Basic Functions and their Domain and Range. Odd, Even Functions, and their Symmetry.. Translations

More information

Text: A Graphical Approach to College Algebra (Hornsby, Lial, Rockswold)

Text: A Graphical Approach to College Algebra (Hornsby, Lial, Rockswold) Students will take Self Tests covering the topics found in Chapter R (Reference: Basic Algebraic Concepts) and Chapter 1 (Linear Functions, Equations, and Inequalities). If any deficiencies are revealed,

More information

Section 1. Inequalities -5-4 -3-2 -1 0 1 2 3 4 5

Section 1. Inequalities -5-4 -3-2 -1 0 1 2 3 4 5 Worksheet 2.4 Introduction to Inequalities Section 1 Inequalities The sign < stands for less than. It was introduced so that we could write in shorthand things like 3 is less than 5. This becomes 3 < 5.

More information

Math 120 Final Exam Practice Problems, Form: A

Math 120 Final Exam Practice Problems, Form: A Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,

More information

Algebra 2: Q1 & Q2 Review

Algebra 2: Q1 & Q2 Review Name: Class: Date: ID: A Algebra 2: Q1 & Q2 Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. Which is the graph of y = 2(x 2) 2 4? a. c. b. d. Short

More information

Solving Inequalities Examples

Solving Inequalities Examples Solving Inequalities Examples 1. Joe and Katie are dancers. Suppose you compare their weights. You can make only one of the following statements. Joe s weight is less than Kate s weight. Joe s weight is

More information

Name: where Nx ( ) and Dx ( ) are the numerator and

Name: where Nx ( ) and Dx ( ) are the numerator and Oblique and Non-linear Asymptote Activity Name: Prior Learning Reminder: Rational Functions In the past we discussed vertical and horizontal asymptotes of the graph of a rational function of the form m

More information

Date: Section P.2: Exponents and Radicals. Properties of Exponents: Example #1: Simplify. a.) 3 4. b.) 2. c.) 3 4. d.) Example #2: Simplify. b.) a.

Date: Section P.2: Exponents and Radicals. Properties of Exponents: Example #1: Simplify. a.) 3 4. b.) 2. c.) 3 4. d.) Example #2: Simplify. b.) a. Properties of Exponents: Section P.2: Exponents and Radicals Date: Example #1: Simplify. a.) 3 4 b.) 2 c.) 34 d.) Example #2: Simplify. a.) b.) c.) d.) 1 Square Root: Principal n th Root: Example #3: Simplify.

More information

ALGEBRA I A PLUS COURSE OUTLINE

ALGEBRA I A PLUS COURSE OUTLINE ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best

More information

Section 2.7 One-to-One Functions and Their Inverses

Section 2.7 One-to-One Functions and Their Inverses Section. One-to-One Functions and Their Inverses One-to-One Functions HORIZONTAL LINE TEST: A function is one-to-one if and only if no horizontal line intersects its graph more than once. EXAMPLES: 1.

More information

Chapter 7 - Roots, Radicals, and Complex Numbers

Chapter 7 - Roots, Radicals, and Complex Numbers Math 233 - Spring 2009 Chapter 7 - Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the

More information

Table of Contents Sequence List

Table of Contents Sequence List Table of Contents Sequence List 368-102215 Level 1 Level 5 1 A1 Numbers 0-10 63 H1 Algebraic Expressions 2 A2 Comparing Numbers 0-10 64 H2 Operations and Properties 3 A3 Addition 0-10 65 H3 Evaluating

More information