3.5. Solving Inequalities. Introduction. Prerequisites. Learning Outcomes

Save this PDF as:

Size: px
Start display at page:

Transcription

1 Solving Inequalities 3.5 Introduction An inequality is an expression involving one of the symbols,, > or <. This Section will first show how to manipulate inequalities correctly. Then algebraical and graphical methods of solving inequalities will be described. Prerequisites Before starting this Section you should... Learning Outcomes On completion you should be able to... be able to solve linear and quadratic equations re-arrange expressions involving inequalities solve linear and quadratic inequalities 50 HELM (2008): Workbook 3: Equations, Inequalities & Partial Fractions

2 1. The inequality symbols Recall the definitions of the inequality symbols in Key Point 11: Key Point 11 The symbols >, <,, are called inequalities > means: is greater than, means: is greater than or equal to < means: is less than, means: is less than or equal to So for example, 8 > < A number line is often a helpful way of picturing inequalities. Given two numbers a and b, if b > a then b will be to the right of a on the number line as shown in Figure 9. a b Figure 9: When b > a, b is to the right of a on the number line. Note from Figure 10 that 3 > 5, 4 > 2 and 8 > Figure 10 Inequalities can always be written in two ways. For example in English we can state that 8 is greater than 7, or equivalently, that 7 is less than 8. Mathematically we write 8 > 7 or 7 < 8. In general if b > a then a < b. If a < b then a will be to the left of b on the number line. Example 33 Rewrite the inequality 2 5 < x using only the greater than sign, >. 2 5 < x can be written as x > 2 5 HELM (2008): Section 3.5: Solving Inequalities 51

3 Example 34 Rewrite the inequality 5 > x using only the less than sign, <. 5 > x can be written as x < 5. Sometimes two inequalities are combined into a single statement. Consider for example the statement 3 < x < 6. This is a compact way of writing 3 < x and x < 6. Now 3 < x is equivalent to x > 3 and so 3 < x < 6 means x is greater than 3 but less than 6. Inequalities obey simple rules when used in conjunction with arithmetical operations: Key Point Adding or subtracting the same quantity from both sides of an inequality leaves the inequality symbol unchanged. 2. Multiplying or dividing both sides by a positive number leaves the inequality unchanged. 3. Multiplying or dividing both sides by a negative number reverses the inequality. For example, since 8 > 5, by adding k to both sides we can state 8 + k > 5 + k for any value of k. For example (with k = 3) 8 3 > 5 3. Further, by multiplying both sides of 8 > 5 by k we can state 8k > 5k provided k is positive. However, 8k < 5k if k is negative. We emphasise that the inequality sign is reversed when multiplying both sides by a negative number. A common mistake is to forget to reverse the inequality symbol. For example if 8 > 5, multiplying both sides by 1 gives 8 < HELM (2008): Workbook 3: Equations, Inequalities & Partial Fractions

4 Task Find the result of multiplying both sides of the inequality 18 < 9 by > 27 The modulus or magnitude sign is sometimes used with inequalities. For example x < 1 represents the set of all numbers whose actual size, irrespective of sign, is less than 1. This means any value between 1 and 1. Thus x < 1 means 1 < x < 1 Similarly x > 4 means all numbers whose size, irrespective of sign, is greater than 4. This means any value greater than 4 or less than 4. Thus x > 4 means x > 4 or x < 4 In general, if k is a positive number: Key Point 13 x < k means k < x < k x > k means x > k or x < k Exercises 1. State which of the following statements are true and which are false. (a) 4 > 9, (b) 4 > 4, (c) 4 4, (d) < 10 5, (e) 19 < 100, (f) 19 > 20, (g) In questions 2-9 rewrite each of the statements without using a modulus sign: 2. x < 2, 3. x < 5, 4. x 7.5, 5. x 3 < 2, 6. x a < 1, 7. x > 2, 8. x > 7.5, 9. x 0. HELM (2008): Section 3.5: Solving Inequalities 53

5 s 1. (a) F (b) F (c) T (d) F (e) T (f) T (g) T 2. 2 < x < < x < x < x 3 < < x a < 1 7. x > 2 or x < 2 8. x > 7.5 or x < x 0 or x 0, in fact any x. 2. Solving linear inequalities algebraically When we are asked to solve an inequality, the inequality will contain an unknown variable, say x. Solving means obtaining all values of x for which the inequality is true. In a linear inequality the unknown appears only to the first power, that is as x, and not as x 2, x 3, x 1/2 and so on. Consider the following examples. Example 35 Solve the inequality 4x + 3 > 0. 4x + 3 > 0 4x > 3, by subtracting 3 from both sides x > 3 4 by dividing both sides by 4. Hence all values of x greater than 3 4 satisfy 4x + 3 > 0. Example 36 Solve the inequality 3x x 7 0 3x 7 by adding 7 to both sides x 7 3 dividing both sides by 3 and reversing the inequality Hence all values of x greater than or equal to 7 3 satisfy 3x HELM (2008): Workbook 3: Equations, Inequalities & Partial Fractions

6 Task Solve the inequality 17x + 2 < 4x + 1. This is done by making x the subject and obtain it on its own on the left-hand side. Start by subtracting 4x from both sides to remove quantities involving x from the right: 13x + 2 < 1 Now subtract 2 from both sides to remove the 2 on the left: 13x < 1. Finally, the range of values of x are x < 1/13 Example 37 Solve the inequality 5x 2 < 4 and depict the solution graphically. 5x 2 < 4 is equivalent to 4 < 5x 2 < 4 We treat each part of the inequality separately: 4 < 5x 2 2 < 5x by adding 2 to both sides 2 5 < x by dividing both sides by 5 So x > 2. Now consider the second part: 5x 2 < x 2 < 4 5x < 6 by adding 2 to both sides x < 6 5 by dividing both sides by 5 So x < 6 5. HELM (2008): Section 3.5: Solving Inequalities 55

7 (contd.) Putting both parts of the solution together we see that the inequality is satisfied when 2 < x < 6. This range of values is shown in Figure /5 0 6/5 Figure 11: 5x 2 < 4 which is equivalent to 2 5 < x < 6 5 Task Solve the inequality 1 2x < 5. First of all rewrite the inequality without using the modulus sign: 1 2x < 5 is equivalent to: 5 < 1 2x < 5 Then treat each part separately. First of all consider 5 < 1 2x. Solve this: x < 3 The second part is 1 2x < 5. Solve this. x > 2 Finally, give the solution as one statement: 2 < x < HELM (2008): Workbook 3: Equations, Inequalities & Partial Fractions

8 Exercises In the following questions solve the given inequality algebraically. 1. 4x > x > x > x x > x < x > x > x < x x > x x x x x + 1 < x x x + 2 > x + 11 > x x x < x > 2x x 3 > x x < x x > x 3 s 1. x > 2 2. x > 8/5 3. x > 5/8 4. x 5/8 5. x > 1/2 6. x < 1/3 7. x > 2/5 8. x > 0 9. x < x x > 4/3 12. x > 4/3 13. x 3/4 14. x 4/3 15. x x x < 7/5 18. x 7/5 19. x 3/7 20. x > 7/ x > 11/ x 1/5 23. x 1/5 24. x > 9/7 25. x > x > 4/7 or x < 2/7 27. x 1 or x /5 < x < 1/5 29. x = x < 1/5, x > 3/5 31. x 1/5, x 1 3. Solving inequalities using graphs Graphs can be used to help solve inequalities. This approach is particularly useful if the inequality is not linear as, in these cases solving the inequalities algebraically can often be very tricky. Graphics calculators or software can save a lot of time and effort here. Example 38 Solve graphically the inequality 5x + 2 < y y = 5x + 2 x = 2/ x Figure 12: Graph of y = 5x + 2. We consider the function y = 5x + 2 whose graph is shown in Figure 12. The values of x which make 5x + 2 negative are those for which y is negative. We see directly from the graph that y is negative when x < 2 5. HELM (2008): Section 3.5: Solving Inequalities 57

9 Example 39 Find the range of values of x for which x 2 x 6 < 0. We consider the graph of y = x 2 x 6 which is shown in Figure 13. y y = x 2 x 6 x 5 Figure 13: Graph of y = x 2 x 6 Note that the graph crosses the x axis when x = 2 and when x = 3, and x 2 x 6 will be negative when y is negative. Directly from the graph we see that y is negative when 2 < x < 3. Task Find the range of values of x for which x 2 x 6 > 0. The graph of y = x 2 x 6 has been drawn in Figure 13. We require y = x 2 x 6 to be positive. Use the graph to solve the problem: x < 2 or x > 3 58 HELM (2008): Workbook 3: Equations, Inequalities & Partial Fractions

10 Example 40 By plotting a graph of y = 20x 4 4x 3 143x x find the range of values of x for which 20x 4 4x 3 143x x < 0 A software package has been used to plot the graph which is shown in Figure 14. We see that y is negative when 2.5 < x < 1 and is also negative when 1.5 < x < 2.2. y x Figure 14: Graph of y = 20x 4 4x 3 143x x Exercises In questions 1-5 solve the given inequality graphically: 1. 3x + 1 < x 7 < x + 9 > 0, 4. 5x 3 > 0 5. x 2 x 6 < 0 s 1. x < 1/3 2. x < 7/2, 3. x > 3/2 4. x > 3/ < x < 3 HELM (2008): Section 3.5: Solving Inequalities 59

3.6. Partial Fractions. Introduction. Prerequisites. Learning Outcomes

Partial Fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. For 4x + 7 example it can be shown that x 2 + 3x + 2 has the same

Equations, Inequalities & Partial Fractions

Contents Equations, Inequalities & Partial Fractions.1 Solving Linear Equations 2.2 Solving Quadratic Equations 1. Solving Polynomial Equations 1.4 Solving Simultaneous Linear Equations 42.5 Solving Inequalities

3.2. Solving quadratic equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving quadratic equations 3.2 Introduction A quadratic equation is one which can be written in the form ax 2 + bx + c = 0 where a, b and c are numbers and x is the unknown whose value(s) we wish to find.

is identically equal to x 2 +3x +2

Partial fractions.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as + for any

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

is identically equal to x 2 +3x +2

Partial fractions 3.6 Introduction It is often helpful to break down a complicated algebraic fraction into a sum of simpler fractions. 4x+7 For example it can be shown that has the same value as 1 + 3

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

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

Temperature Scales. The metric system that we are now using includes a unit that is specific for the representation of measured temperatures.

Temperature Scales INTRODUCTION The metric system that we are now using includes a unit that is specific for the representation of measured temperatures. The unit of temperature in the metric system is

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

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

Solutions of Linear Equations in One Variable

2. Solutions of Linear Equations in One Variable 2. OBJECTIVES. Identify a linear equation 2. Combine like terms to solve an equation We begin this chapter by considering one of the most important tools

Section 5.4 The Quadratic Formula 481 5.4 The Quadratic Formula Consider the general quadratic function f(x) = ax + bx + c. In the previous section, we learned that we can find the zeros of this function

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.

1.4. Arithmetic of Algebraic Fractions. Introduction. Prerequisites. Learning Outcomes

Arithmetic of Algebraic Fractions 1.4 Introduction Just as one whole number divided by another is called a numerical fraction, so one algebraic expression divided by another is known as an algebraic fraction.

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

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

0.8 Rational Expressions and Equations

96 Prerequisites 0.8 Rational Expressions and Equations We now turn our attention to rational expressions - that is, algebraic fractions - and equations which contain them. The reader is encouraged to

2 Session Two - Complex Numbers and Vectors

PH2011 Physics 2A Maths Revision - Session 2: Complex Numbers and Vectors 1 2 Session Two - Complex Numbers and Vectors 2.1 What is a Complex Number? The material on complex numbers should be familiar

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

Click on the links below to jump directly to the relevant section

Click on the links below to jump directly to the relevant section What is algebra? Operations with algebraic terms Mathematical properties of real numbers Order of operations What is Algebra? Algebra is

Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704.

Basic Math Refresher A tutorial and assessment of basic math skills for students in PUBP704. The purpose of this Basic Math Refresher is to review basic math concepts so that students enrolled in PUBP704:

Arithmetic Operations. The real numbers have the following properties: In particular, putting a 1 in the Distributive Law, we get

Review of Algebra REVIEW OF ALGEBRA Review of Algebra Here we review the basic rules and procedures of algebra that you need to know in order to be successful in calculus. Arithmetic Operations The real

Learning Objectives for Section 1.1 Linear Equations and Inequalities

Learning Objectives for Section 1.1 Linear Equations and Inequalities After this lecture and the assigned homework, you should be able to solve linear equations. solve linear inequalities. use interval

Linear Equations and Inequalities

Linear Equations and Inequalities Section 1.1 Prof. Wodarz Math 109 - Fall 2008 Contents 1 Linear Equations 2 1.1 Standard Form of a Linear Equation................ 2 1.2 Solving Linear Equations......................

Matrix Solution of Equations

Contents 8 Matrix Solution of Equations 8.1 Solution by Cramer s Rule 2 8.2 Solution by Inverse Matrix Method 13 8.3 Solution by Gauss Elimination 22 Learning outcomes In this Workbook you will learn to

Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving

Section 7 Algebraic Manipulations and Solving Part 1 Expressions, Equations, and Inequalities: Simplifying and Solving Before launching into the mathematics, let s take a moment to talk about the words

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

3.1. Solving linear equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Solving linear equations 3.1 Introduction Many problems in engineering reduce to the solution of an equation or a set of equations. An equation is a type of mathematical expression which contains one or

Review of Fundamental Mathematics

Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

Determine If An Equation Represents a Function

Question : What is a linear function? The term linear function consists of two parts: linear and function. To understand what these terms mean together, we must first understand what a function is. The

1.4 Compound Inequalities

Section 1.4 Compound Inequalities 53 1.4 Compound Inequalities This section discusses a technique that is used to solve compound inequalities, which is a phrase that usually refers to a pair of inequalities

LINEAR INEQUALITIES. Mathematics is the art of saying many things in many different ways. MAXWELL

Chapter 6 LINEAR INEQUALITIES 6.1 Introduction Mathematics is the art of saying many things in many different ways. MAXWELL In earlier classes, we have studied equations in one variable and two variables

Balancing Chemical Equations

Balancing Chemical Equations A mathematical equation is simply a sentence that states that two expressions are equal. One or both of the expressions will contain a variable whose value must be determined

3.1. RATIONAL EXPRESSIONS

3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers

7.7 Solving Rational Equations

Section 7.7 Solving Rational Equations 7 7.7 Solving Rational Equations When simplifying comple fractions in the previous section, we saw that multiplying both numerator and denominator by the appropriate

Properties of Real Numbers

16 Chapter P Prerequisites P.2 Properties of Real Numbers What you should learn: Identify and use the basic properties of real numbers Develop and use additional properties of real numbers Why you should

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

Accuplacer Arithmetic Study Guide

Testing Center Student Success Center Accuplacer Arithmetic Study Guide I. Terms Numerator: which tells how many parts you have (the number on top) Denominator: which tells how many parts in the whole

Algebra Chapter 6 Notes Systems of Equations and Inequalities. Lesson 6.1 Solve Linear Systems by Graphing System of linear equations:

Algebra Chapter 6 Notes Systems of Equations and Inequalities Lesson 6.1 Solve Linear Systems by Graphing System of linear equations: Solution of a system of linear equations: Consistent independent system:

3.3 Addition and Subtraction of Rational Numbers

3.3 Addition and Subtraction of Rational Numbers In this section we consider addition and subtraction of both fractions and decimals. We start with addition and subtraction of fractions with the same denominator.

1.6 The Order of Operations

1.6 The Order of Operations Contents: Operations Grouping Symbols The Order of Operations Exponents and Negative Numbers Negative Square Roots Square Root of a Negative Number Order of Operations and Negative

Actually, if you have a graphing calculator this technique can be used to find solutions to any equation, not just quadratics. All you need to do is

QUADRATIC EQUATIONS Definition ax 2 + bx + c = 0 a, b, c are constants (generally integers) Roots Synonyms: Solutions or Zeros Can have 0, 1, or 2 real roots Consider the graph of quadratic equations.

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

Algebraic expressions are a combination of numbers and variables. Here are examples of some basic algebraic expressions.

Page 1 of 13 Review of Linear Expressions and Equations Skills involving linear equations can be divided into the following groups: Simplifying algebraic expressions. Linear expressions. Solving linear

Tom wants to find two real numbers, a and b, that have a sum of 10 and have a product of 10. He makes this table.

Sum and Product This problem gives you the chance to: use arithmetic and algebra to represent and analyze a mathematical situation solve a quadratic equation by trial and improvement Tom wants to find

Section 1.4 Graphs of Linear Inequalities

Section 1.4 Graphs of Linear Inequalities A Linear Inequality and its Graph A linear inequality has the same form as a linear equation, except that the equal symbol is replaced with any one of,,

8.2. Solution by Inverse Matrix Method. Introduction. Prerequisites. Learning Outcomes

Solution by Inverse Matrix Method 8.2 Introduction The power of matrix algebra is seen in the representation of a system of simultaneous linear equations as a matrix equation. Matrix algebra allows us

( yields. Combining the terms in the numerator you arrive at the answer:

Algebra Skillbuilder Solutions: 1. Starting with, you ll need to find a common denominator to add/subtract the fractions. If you choose the common denominator 15, you can multiply each fraction by one

Biggar High School Mathematics Department. National 5 Learning Intentions & Success Criteria: Assessing My Progress

Biggar High School Mathematics Department National 5 Learning Intentions & Success Criteria: Assessing My Progress Expressions & Formulae Topic Learning Intention Success Criteria I understand this Approximation

GCSE MATHEMATICS. 43602H Unit 2: Number and Algebra (Higher) Report on the Examination. Specification 4360 November 2014. Version: 1.

GCSE MATHEMATICS 43602H Unit 2: Number and Algebra (Higher) Report on the Examination Specification 4360 November 2014 Version: 1.0 Further copies of this Report are available from aqa.org.uk Copyright

Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

In this chapter, you will learn to use cost-volume-profit analysis.

2.0 Chapter Introduction In this chapter, you will learn to use cost-volume-profit analysis. Assumptions. When you acquire supplies or services, you normally expect to pay a smaller price per unit as the

ALGEBRA. sequence, term, nth term, consecutive, rule, relationship, generate, predict, continue increase, decrease finite, infinite

ALGEBRA Pupils should be taught to: Generate and describe sequences As outcomes, Year 7 pupils should, for example: Use, read and write, spelling correctly: sequence, term, nth term, consecutive, rule,

Florida Department of Education/Office of Assessment January 2012. Algebra 1 End-of-Course Assessment Achievement Level Descriptions

Florida Department of Education/Office of Assessment January 2012 Algebra 1 End-of-Course Assessment Achievement Level Descriptions Algebra 1 EOC Assessment Reporting Category Functions, Linear Equations,

Year 9 set 1 Mathematics notes, to accompany the 9H book.

Part 1: Year 9 set 1 Mathematics notes, to accompany the 9H book. equations 1. (p.1), 1.6 (p. 44), 4.6 (p.196) sequences 3. (p.115) Pupils use the Elmwood Press Essential Maths book by David Raymer (9H

Graphical Presentation of Data

Graphical Presentation of Data Guidelines for Making Graphs Titles should tell the reader exactly what is graphed Remove stray lines, legends, points, and any other unintended additions by the computer

Variable. 1.1 Order of Operations. August 17, evaluating expressions ink.notebook. Standards. letter or symbol used to represent a number

1.1 evaluating expressions ink.notebook page 8 Unit 1 Basic Equations and Inequalities 1.1 Order of Operations page 9 Square Cube Variable Variable Expression Exponent page 10 page 11 1 Lesson Objectives

Mathematical goals. Starting points. Materials required. Time needed

Level A3 of challenge: C A3 Creating and solving harder equations equations Mathematical goals Starting points Materials required Time needed To enable learners to: create and solve equations, where the

Chapter 8 Graphs and Functions:

Chapter 8 Graphs and Functions: Cartesian axes, coordinates and points 8.1 Pictorially we plot points and graphs in a plane (flat space) using a set of Cartesian axes traditionally called the x and y axes

01 In any business, or, indeed, in life in general, hindsight is a beautiful thing. If only we could look into a

01 technical cost-volumeprofit relevant to acca qualification paper F5 In any business, or, indeed, in life in general, hindsight is a beautiful thing. If only we could look into a crystal ball and find

COLLEGE ALGEBRA 10 TH EDITION LIAL HORNSBY SCHNEIDER 1.1-1

10 TH EDITION COLLEGE ALGEBRA LIAL HORNSBY SCHNEIDER 1.1-1 1.1 Linear Equations Basic Terminology of Equations Solving Linear Equations Identities 1.1-2 Equations An equation is a statement that two expressions

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

Equations and Inequalities

Rational Equations Overview of Objectives, students should be able to: 1. Solve rational equations with variables in the denominators.. Recognize identities, conditional equations, and inconsistent equations.

TYPES OF NUMBERS. Example 2. Example 1. Problems. Answers

TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have exactly two factors, namely, one and itself, are called prime

Equations, Inequalities, Solving. and Problem AN APPLICATION

Equations, Inequalities, and Problem Solving. Solving Equations. Using the Principles Together AN APPLICATION To cater a party, Curtis Barbeque charges a \$0 setup fee plus \$ per person. The cost of Hotel

MATH 60 NOTEBOOK CERTIFICATIONS

MATH 60 NOTEBOOK CERTIFICATIONS Chapter #1: Integers and Real Numbers 1.1a 1.1b 1.2 1.3 1.4 1.8 Chapter #2: Algebraic Expressions, Linear Equations, and Applications 2.1a 2.1b 2.1c 2.2 2.3a 2.3b 2.4 2.5

Solving Systems of Two Equations Algebraically

8 MODULE 3. EQUATIONS 3b Solving Systems of Two Equations Algebraically Solving Systems by Substitution In this section we introduce an algebraic technique for solving systems of two equations in two unknowns

Algebra I Notes Relations and Functions Unit 03a

OBJECTIVES: F.IF.A.1 Understand the concept of a function and use function notation. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element

1 Determine whether an. 2 Solve systems of linear. 3 Solve systems of linear. 4 Solve systems of linear. 5 Select the most efficient

Section 3.1 Systems of Linear Equations in Two Variables 163 SECTION 3.1 SYSTEMS OF LINEAR EQUATIONS IN TWO VARIABLES Objectives 1 Determine whether an ordered pair is a solution of a system of linear

Solving Linear Equations - General Equations

1.3 Solving Linear Equations - General Equations Objective: Solve general linear equations with variables on both sides. Often as we are solving linear equations we will need to do some work to set them

Indicator 2: Use a variety of algebraic concepts and methods to solve equations and inequalities.

3 rd Grade Math Learning Targets Algebra: Indicator 1: Use procedures to transform algebraic expressions. 3.A.1.1. Students are able to explain the relationship between repeated addition and multiplication.

Solving simultaneous equations using the inverse matrix

Solving simultaneous equations using the inverse matrix 8.2 Introduction The power of matrix algebra is seen in the representation of a system of simultaneous linear equations as a matrix equation. Matrix

MBA Jump Start Program

MBA Jump Start Program Module 2: Mathematics Thomas Gilbert Mathematics Module Online Appendix: Basic Mathematical Concepts 2 1 The Number Spectrum Generally we depict numbers increasing from left to right

Let s explore the content and skills assessed by Heart of Algebra questions.

Chapter 9 Heart of Algebra Heart of Algebra focuses on the mastery of linear equations, systems of linear equations, and linear functions. The ability to analyze and create linear equations, inequalities,

ModuMath Algebra Lessons

ModuMath Algebra Lessons Program Title 1 Getting Acquainted With Algebra 2 Order of Operations 3 Adding & Subtracting Algebraic Expressions 4 Multiplying Polynomials 5 Laws of Algebra 6 Solving Equations

Section 2.1 Intercepts; Symmetry; Graphing Key Equations

Intercepts: An intercept is the point at which a graph crosses or touches the coordinate axes. x intercept is 1. The point where the line crosses (or intercepts) the x-axis. 2. The x-coordinate of a point

This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.

Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course

The Graphical Method: An Example

The Graphical Method: An Example Consider the following linear program: Maximize 4x 1 +3x 2 Subject to: 2x 1 +3x 2 6 (1) 3x 1 +2x 2 3 (2) 2x 2 5 (3) 2x 1 +x 2 4 (4) x 1, x 2 0, where, for ease of reference,

POLYNOMIAL FUNCTIONS

POLYNOMIAL FUNCTIONS Polynomial Division.. 314 The Rational Zero Test.....317 Descarte s Rule of Signs... 319 The Remainder Theorem.....31 Finding all Zeros of a Polynomial Function.......33 Writing a

Mathematics Online Instructional Materials Correlation to the 2009 Algebra I Standards of Learning and Curriculum Framework

Provider York County School Division Course Syllabus URL http://yorkcountyschools.org/virtuallearning/coursecatalog.aspx Course Title Algebra I AB Last Updated 2010 - A.1 The student will represent verbal

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

Linear Programming. Solving LP Models Using MS Excel, 18

SUPPLEMENT TO CHAPTER SIX Linear Programming SUPPLEMENT OUTLINE Introduction, 2 Linear Programming Models, 2 Model Formulation, 4 Graphical Linear Programming, 5 Outline of Graphical Procedure, 5 Plotting

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

This assignment will help you to prepare for Algebra 1 by reviewing some of the things you learned in Middle School. If you cannot remember how to complete a specific problem, there is an example at the

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

8.1. Cramer s Rule for Solving Simultaneous Linear Equations. Introduction. Prerequisites. Learning Outcomes. Learning Style

Cramer s Rule for Solving Simultaneous Linear Equations 8.1 Introduction The need to solve systems of linear equations arises frequently in engineering. The analysis of electric circuits and the control

Sequences. A sequence is a list of numbers, or a pattern, which obeys a rule.

Sequences A sequence is a list of numbers, or a pattern, which obeys a rule. Each number in a sequence is called a term. ie the fourth term of the sequence 2, 4, 6, 8, 10, 12... is 8, because it is the

Math Review. for the Quantitative Reasoning Measure of the GRE revised General Test

Math Review for the Quantitative Reasoning Measure of the GRE revised General Test www.ets.org Overview This Math Review will familiarize you with the mathematical skills and concepts that are important

Algebra I Credit Recovery

Algebra I Credit Recovery COURSE DESCRIPTION: The purpose of this course is to allow the student to gain mastery in working with and evaluating mathematical expressions, equations, graphs, and other topics,

EQUATIONS and INEQUALITIES

EQUATIONS and INEQUALITIES Linear Equations and Slope 1. Slope a. Calculate the slope of a line given two points b. Calculate the slope of a line parallel to a given line. c. Calculate the slope of a line

MATHEMATICS FOR ENGINEERING BASIC ALGEBRA

MATHEMATICS FOR ENGINEERING BASIC ALGEBRA TUTORIAL 3 EQUATIONS This is the one of a series of basic tutorials in mathematics aimed at beginners or anyone wanting to refresh themselves on fundamentals.

Math Common Core Sampler Test

High School Algebra Core Curriculum Math Test Math Common Core Sampler Test Our High School Algebra sampler covers the twenty most common questions that we see targeted for this level. For complete tests

LINEAR EQUATIONS IN TWO VARIABLES

66 MATHEMATICS CHAPTER 4 LINEAR EQUATIONS IN TWO VARIABLES The principal use of the Analytic Art is to bring Mathematical Problems to Equations and to exhibit those Equations in the most simple terms that

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

Math 0980 Chapter Objectives. Chapter 1: Introduction to Algebra: The Integers.

Math 0980 Chapter Objectives Chapter 1: Introduction to Algebra: The Integers. 1. Identify the place value of a digit. 2. Write a number in words or digits. 3. Write positive and negative numbers used

Core Maths C1. Revision Notes

Core Maths C Revision Notes November 0 Core Maths C Algebra... Indices... Rules of indices... Surds... 4 Simplifying surds... 4 Rationalising the denominator... 4 Quadratic functions... 4 Completing the